Dirichlet series
| L(s) = 1 | + 6·5-s + 2·7-s + 18·17-s − 16·23-s + 21·25-s − 12·31-s + 12·35-s − 14·37-s + 28·43-s − 6·47-s + 2·49-s − 10·53-s + 60·61-s + 8·67-s − 8·71-s + 78·73-s − 6·81-s + 108·85-s − 48·101-s − 84·103-s − 40·107-s − 8·113-s − 96·115-s + 36·119-s + 62·121-s + 54·125-s + 127-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 0.755·7-s + 4.36·17-s − 3.33·23-s + 21/5·25-s − 2.15·31-s + 2.02·35-s − 2.30·37-s + 4.26·43-s − 0.875·47-s + 2/7·49-s − 1.37·53-s + 7.68·61-s + 0.977·67-s − 0.949·71-s + 9.12·73-s − 2/3·81-s + 11.7·85-s − 4.77·101-s − 8.27·103-s − 3.86·107-s − 0.752·113-s − 8.95·115-s + 3.30·119-s + 5.63·121-s + 4.82·125-s + 0.0887·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.94952\) |
| Root analytic conductor: | \(1.05731\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.068600068\) |
| \(L(\frac12)\) | \(\approx\) | \(4.068600068\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 6 T + 3 p T^{2} - 18 T^{3} - 3 T^{4} + 36 T^{5} - 78 T^{6} + 336 T^{7} - 1006 T^{8} + 336 p T^{9} - 78 p^{2} T^{10} + 36 p^{3} T^{11} - 3 p^{4} T^{12} - 18 p^{5} T^{13} + 3 p^{7} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 2 T + 2 T^{2} + 16 T^{3} - 83 T^{4} + 4 p^{2} T^{5} - 2 p^{2} T^{6} - 30 p^{2} T^{7} + 132 p^{2} T^{8} - 30 p^{3} T^{9} - 2 p^{4} T^{10} + 4 p^{5} T^{11} - 83 p^{4} T^{12} + 16 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( 1 + 2 p T^{4} + 4 p T^{5} + 28 p T^{7} - 31 T^{8} + 8 p^{2} T^{9} + 8 p^{2} T^{10} + 256 p T^{11} + 2 p^{3} T^{12} - 92 p^{2} T^{13} + 440 p^{2} T^{14} - 244 p T^{15} + 7780 T^{16} - 244 p^{2} T^{17} + 440 p^{4} T^{18} - 92 p^{5} T^{19} + 2 p^{7} T^{20} + 256 p^{6} T^{21} + 8 p^{8} T^{22} + 8 p^{9} T^{23} - 31 p^{8} T^{24} + 28 p^{10} T^{25} + 4 p^{12} T^{27} + 2 p^{13} T^{28} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 31 T^{2} - 12 T^{3} + 511 T^{4} + 252 T^{5} - 6412 T^{6} - 1392 T^{7} + 71182 T^{8} - 1392 p T^{9} - 6412 p^{2} T^{10} + 252 p^{3} T^{11} + 511 p^{4} T^{12} - 12 p^{5} T^{13} - 31 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 - 384 T^{4} + 95932 T^{8} - 12871296 T^{12} + 2070861510 T^{16} - 12871296 p^{4} T^{20} + 95932 p^{8} T^{24} - 384 p^{12} T^{28} + p^{16} T^{32} \) | |
| 17 | \( 1 - 18 T + 162 T^{2} - 972 T^{3} + 3843 T^{4} - 5328 T^{5} - 54270 T^{6} + 521298 T^{7} - 2465507 T^{8} + 5624676 T^{9} + 14219712 T^{10} - 209564892 T^{11} + 1084680234 T^{12} - 2931912576 T^{13} - 1788585948 T^{14} + 64173985380 T^{15} - 361145427270 T^{16} + 64173985380 p T^{17} - 1788585948 p^{2} T^{18} - 2931912576 p^{3} T^{19} + 1084680234 p^{4} T^{20} - 209564892 p^{5} T^{21} + 14219712 p^{6} T^{22} + 5624676 p^{7} T^{23} - 2465507 p^{8} T^{24} + 521298 p^{9} T^{25} - 54270 p^{10} T^{26} - 5328 p^{11} T^{27} + 3843 p^{12} T^{28} - 972 p^{13} T^{29} + 162 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 78 T^{2} + 2559 T^{4} - 64506 T^{6} + 1886917 T^{8} - 45195540 T^{10} + 809434458 T^{12} - 895779768 p T^{14} + 374666887482 T^{16} - 895779768 p^{3} T^{18} + 809434458 p^{4} T^{20} - 45195540 p^{6} T^{22} + 1886917 p^{8} T^{24} - 64506 p^{10} T^{26} + 2559 p^{12} T^{28} - 78 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 16 T + 128 T^{2} + 232 T^{3} - 4650 T^{4} - 52804 T^{5} - 222752 T^{6} + 287076 T^{7} + 10138289 T^{8} + 62811488 T^{9} + 132218856 T^{10} - 911730824 T^{11} - 9742362906 T^{12} - 41221272300 T^{13} - 26389121736 T^{14} + 830176620924 T^{15} + 5986766744932 T^{16} + 830176620924 p T^{17} - 26389121736 p^{2} T^{18} - 41221272300 p^{3} T^{19} - 9742362906 p^{4} T^{20} - 911730824 p^{5} T^{21} + 132218856 p^{6} T^{22} + 62811488 p^{7} T^{23} + 10138289 p^{8} T^{24} + 287076 p^{9} T^{25} - 222752 p^{10} T^{26} - 52804 p^{11} T^{27} - 4650 p^{12} T^{28} + 232 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 70 T^{2} + 2433 T^{4} - 88862 T^{6} + 3077972 T^{8} - 88862 p^{2} T^{10} + 2433 p^{4} T^{12} - 70 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 6 T + 75 T^{2} + 378 T^{3} + 2289 T^{4} + 15168 T^{5} + 89982 T^{6} + 703668 T^{7} + 3775322 T^{8} + 703668 p T^{9} + 89982 p^{2} T^{10} + 15168 p^{3} T^{11} + 2289 p^{4} T^{12} + 378 p^{5} T^{13} + 75 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 14 T + 98 T^{2} + 108 T^{3} - 5821 T^{4} - 62872 T^{5} - 6 p^{3} T^{6} + 264482 T^{7} + 19235053 T^{8} + 155496292 T^{9} + 527814464 T^{10} - 2237395644 T^{11} - 42158420550 T^{12} - 239649719768 T^{13} - 288310266268 T^{14} + 6869428795396 T^{15} + 67515487693706 T^{16} + 6869428795396 p T^{17} - 288310266268 p^{2} T^{18} - 239649719768 p^{3} T^{19} - 42158420550 p^{4} T^{20} - 2237395644 p^{5} T^{21} + 527814464 p^{6} T^{22} + 155496292 p^{7} T^{23} + 19235053 p^{8} T^{24} + 264482 p^{9} T^{25} - 6 p^{13} T^{26} - 62872 p^{11} T^{27} - 5821 p^{12} T^{28} + 108 p^{13} T^{29} + 98 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 158 T^{2} + 12249 T^{4} - 702598 T^{6} + 32571956 T^{8} - 702598 p^{2} T^{10} + 12249 p^{4} T^{12} - 158 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 14 T + 98 T^{2} - 1120 T^{3} + 6485 T^{4} + 5180 T^{5} - 80850 T^{6} + 1706838 T^{7} - 22979132 T^{8} + 1706838 p T^{9} - 80850 p^{2} T^{10} + 5180 p^{3} T^{11} + 6485 p^{4} T^{12} - 1120 p^{5} T^{13} + 98 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 6 T + 18 T^{2} + 36 T^{3} - 2921 T^{4} - 18468 T^{5} - 57582 T^{6} - 252870 T^{7} + 4663233 T^{8} + 33295872 T^{9} + 109070208 T^{10} + 1603653504 T^{11} + 22768480382 T^{12} + 42197450964 T^{13} - 94740600420 T^{14} - 4237289839752 T^{15} - 69054970682542 T^{16} - 4237289839752 p T^{17} - 94740600420 p^{2} T^{18} + 42197450964 p^{3} T^{19} + 22768480382 p^{4} T^{20} + 1603653504 p^{5} T^{21} + 109070208 p^{6} T^{22} + 33295872 p^{7} T^{23} + 4663233 p^{8} T^{24} - 252870 p^{9} T^{25} - 57582 p^{10} T^{26} - 18468 p^{11} T^{27} - 2921 p^{12} T^{28} + 36 p^{13} T^{29} + 18 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 10 T + 50 T^{2} + 188 T^{3} + 23 T^{4} - 11360 T^{5} - 97078 T^{6} - 1940530 T^{7} - 11034527 T^{8} - 98244124 T^{9} - 728776936 T^{10} - 2410860356 T^{11} + 12119827374 T^{12} + 85538769312 T^{13} + 1740588224004 T^{14} + 17220856292660 T^{15} + 182053383396050 T^{16} + 17220856292660 p T^{17} + 1740588224004 p^{2} T^{18} + 85538769312 p^{3} T^{19} + 12119827374 p^{4} T^{20} - 2410860356 p^{5} T^{21} - 728776936 p^{6} T^{22} - 98244124 p^{7} T^{23} - 11034527 p^{8} T^{24} - 1940530 p^{9} T^{25} - 97078 p^{10} T^{26} - 11360 p^{11} T^{27} + 23 p^{12} T^{28} + 188 p^{13} T^{29} + 50 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 206 T^{2} + 19951 T^{4} - 1160698 T^{6} + 36895301 T^{8} + 725030860 T^{10} - 186430156390 T^{12} + 13972955412536 T^{14} - 825805533647366 T^{16} + 13972955412536 p^{2} T^{18} - 186430156390 p^{4} T^{20} + 725030860 p^{6} T^{22} + 36895301 p^{8} T^{24} - 1160698 p^{10} T^{26} + 19951 p^{12} T^{28} - 206 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 30 T + 582 T^{2} - 8460 T^{3} + 99825 T^{4} - 999804 T^{5} + 8996706 T^{6} - 74534946 T^{7} + 592164188 T^{8} - 74534946 p T^{9} + 8996706 p^{2} T^{10} - 999804 p^{3} T^{11} + 99825 p^{4} T^{12} - 8460 p^{5} T^{13} + 582 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 8 T + 32 T^{2} + 808 T^{3} - 16674 T^{4} + 125636 T^{5} - 145088 T^{6} - 9689764 T^{7} + 129576665 T^{8} - 816425320 T^{9} + 28624904 T^{10} + 54970031296 T^{11} - 681977968770 T^{12} + 3748418094196 T^{13} - 97601960616 T^{14} - 251315307372252 T^{15} + 3105056071736884 T^{16} - 251315307372252 p T^{17} - 97601960616 p^{2} T^{18} + 3748418094196 p^{3} T^{19} - 681977968770 p^{4} T^{20} + 54970031296 p^{5} T^{21} + 28624904 p^{6} T^{22} - 816425320 p^{7} T^{23} + 129576665 p^{8} T^{24} - 9689764 p^{9} T^{25} - 145088 p^{10} T^{26} + 125636 p^{11} T^{27} - 16674 p^{12} T^{28} + 808 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 2 T + 2 p T^{2} - 134 T^{3} + 10530 T^{4} - 134 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 - 78 T + 3042 T^{2} - 79092 T^{3} + 1546195 T^{4} - 24359616 T^{5} + 324297090 T^{6} - 3781188906 T^{7} + 39752383773 T^{8} - 386911780068 T^{9} + 3583067122368 T^{10} - 32495729765268 T^{11} + 295688870049386 T^{12} - 2723919940533336 T^{13} + 25146783610706628 T^{14} - 227858270290367460 T^{15} + 1989925377863256890 T^{16} - 227858270290367460 p T^{17} + 25146783610706628 p^{2} T^{18} - 2723919940533336 p^{3} T^{19} + 295688870049386 p^{4} T^{20} - 32495729765268 p^{5} T^{21} + 3583067122368 p^{6} T^{22} - 386911780068 p^{7} T^{23} + 39752383773 p^{8} T^{24} - 3781188906 p^{9} T^{25} + 324297090 p^{10} T^{26} - 24359616 p^{11} T^{27} + 1546195 p^{12} T^{28} - 79092 p^{13} T^{29} + 3042 p^{14} T^{30} - 78 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 362 T^{2} + 68287 T^{4} + 8591326 T^{6} + 767859845 T^{8} + 43076830748 T^{10} + 311275884890 T^{12} - 225684077954216 T^{14} - 26016234382787702 T^{16} - 225684077954216 p^{2} T^{18} + 311275884890 p^{4} T^{20} + 43076830748 p^{6} T^{22} + 767859845 p^{8} T^{24} + 8591326 p^{10} T^{26} + 68287 p^{12} T^{28} + 362 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 + 15582 T^{4} + 51557713 T^{8} - 713327534370 T^{12} - 8636328770393340 T^{16} - 713327534370 p^{4} T^{20} + 51557713 p^{8} T^{24} + 15582 p^{12} T^{28} + p^{16} T^{32} \) | |
| 89 | \( 1 - 528 T^{2} + 148110 T^{4} - 28812000 T^{6} + 4341779857 T^{8} - 540248815104 T^{10} + 58322300299182 T^{12} - 5701133868311568 T^{14} + 521408033078008356 T^{16} - 5701133868311568 p^{2} T^{18} + 58322300299182 p^{4} T^{20} - 540248815104 p^{6} T^{22} + 4341779857 p^{8} T^{24} - 28812000 p^{10} T^{26} + 148110 p^{12} T^{28} - 528 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 + 3704 T^{4} + 114761500 T^{8} - 183344466616 T^{12} + 7721327657662534 T^{16} - 183344466616 p^{4} T^{20} + 114761500 p^{8} T^{24} + 3704 p^{12} T^{28} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.93535818413419187492684644088, −3.84382899481305712348472687809, −3.80797961720774962964075506463, −3.69389968063707294903092063237, −3.60485883141598160916923529719, −3.55731971226450848956354057207, −3.48936879897547103803596212441, −3.20506720343788994783101499864, −3.19246395023333743744005671653, −3.07438739292492255360784098916, −2.94436382440885071260627320325, −2.62168941232299601794351779750, −2.55913576359065439960679677953, −2.48879505025766592904648843031, −2.38318590736236426961807463603, −2.34783080413635529923411753266, −2.12833257748712173246151291087, −2.11132446299984831838171877217, −2.06635450120563584829682270020, −1.79432180712546749212748829072, −1.40374522649104772401200128194, −1.35984856953086130728604036564, −1.23815652034621334078932202121, −1.21951076676920343975389493176, −0.860537810009495429094368963438, 0.860537810009495429094368963438, 1.21951076676920343975389493176, 1.23815652034621334078932202121, 1.35984856953086130728604036564, 1.40374522649104772401200128194, 1.79432180712546749212748829072, 2.06635450120563584829682270020, 2.11132446299984831838171877217, 2.12833257748712173246151291087, 2.34783080413635529923411753266, 2.38318590736236426961807463603, 2.48879505025766592904648843031, 2.55913576359065439960679677953, 2.62168941232299601794351779750, 2.94436382440885071260627320325, 3.07438739292492255360784098916, 3.19246395023333743744005671653, 3.20506720343788994783101499864, 3.48936879897547103803596212441, 3.55731971226450848956354057207, 3.60485883141598160916923529719, 3.69389968063707294903092063237, 3.80797961720774962964075506463, 3.84382899481305712348472687809, 3.93535818413419187492684644088
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.