Dirichlet series
| L(s) = 1 | + 6·7-s − 20·13-s + 8·19-s + 20·25-s − 8·31-s − 4·37-s + 20·43-s + 23·49-s + 28·61-s + 18·67-s − 20·79-s − 120·91-s − 84·97-s + 38·103-s + 24·109-s + 41·121-s + 127-s + 131-s + 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 122·169-s + ⋯ |
| L(s) = 1 | + 2.26·7-s − 5.54·13-s + 1.83·19-s + 4·25-s − 1.43·31-s − 0.657·37-s + 3.04·43-s + 23/7·49-s + 3.58·61-s + 2.19·67-s − 2.25·79-s − 12.5·91-s − 8.52·97-s + 3.74·103-s + 2.29·109-s + 3.72·121-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.38·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{64} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.33882\times 10^{20}\) |
| Root analytic conductor: | \(4.25559\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{64} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.169191980\) |
| \(L(\frac12)\) | \(\approx\) | \(4.169191980\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 - 3 T + 2 T^{2} + 3 T^{3} - 27 T^{4} + 3 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 5 | \( 1 - 4 p T^{2} + 206 T^{4} - 1294 T^{6} + 4787 T^{8} - 3607 T^{10} - 88017 T^{12} + 767823 T^{14} - 4363229 T^{16} + 767823 p^{2} T^{18} - 88017 p^{4} T^{20} - 3607 p^{6} T^{22} + 4787 p^{8} T^{24} - 1294 p^{10} T^{26} + 206 p^{12} T^{28} - 4 p^{15} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 - 41 T^{2} + 785 T^{4} - 7450 T^{6} + 16838 T^{8} + 328499 T^{10} + 1699158 T^{12} - 153375246 T^{14} + 2496017041 T^{16} - 153375246 p^{2} T^{18} + 1699158 p^{4} T^{20} + 328499 p^{6} T^{22} + 16838 p^{8} T^{24} - 7450 p^{10} T^{26} + 785 p^{12} T^{28} - 41 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 5 T + 32 T^{2} + 69 T^{3} + 347 T^{4} + 69 p T^{5} + 32 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 17 | \( 1 - 74 T^{2} + 2453 T^{4} - 57022 T^{6} + 1324580 T^{8} - 32297308 T^{10} + 689971179 T^{12} - 12093908148 T^{14} + 198768259759 T^{16} - 12093908148 p^{2} T^{18} + 689971179 p^{4} T^{20} - 32297308 p^{6} T^{22} + 1324580 p^{8} T^{24} - 57022 p^{10} T^{26} + 2453 p^{12} T^{28} - 74 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 4 T - 36 T^{2} + 262 T^{3} + 503 T^{4} - 6597 T^{5} + 9337 T^{6} + 65501 T^{7} - 346131 T^{8} + 65501 p T^{9} + 9337 p^{2} T^{10} - 6597 p^{3} T^{11} + 503 p^{4} T^{12} + 262 p^{5} T^{13} - 36 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 140 T^{2} + 10514 T^{4} - 545062 T^{6} + 21712127 T^{8} - 707600215 T^{10} + 19840133859 T^{12} - 500942143569 T^{14} + 11826803616955 T^{16} - 500942143569 p^{2} T^{18} + 19840133859 p^{4} T^{20} - 707600215 p^{6} T^{22} + 21712127 p^{8} T^{24} - 545062 p^{10} T^{26} + 10514 p^{12} T^{28} - 140 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 20 T^{2} + 2210 T^{4} + 56736 T^{6} + 2299307 T^{8} + 56736 p^{2} T^{10} + 2210 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 4 T - 91 T^{2} - 232 T^{3} + 5354 T^{4} + 7340 T^{5} - 237291 T^{6} - 82998 T^{7} + 8452027 T^{8} - 82998 p T^{9} - 237291 p^{2} T^{10} + 7340 p^{3} T^{11} + 5354 p^{4} T^{12} - 232 p^{5} T^{13} - 91 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 2 T - 84 T^{2} + 106 T^{3} + 3899 T^{4} - 10725 T^{5} - 92621 T^{6} + 246455 T^{7} + 2134827 T^{8} + 246455 p T^{9} - 92621 p^{2} T^{10} - 10725 p^{3} T^{11} + 3899 p^{4} T^{12} + 106 p^{5} T^{13} - 84 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 217 T^{2} + 23488 T^{4} + 1630195 T^{6} + 79068751 T^{8} + 1630195 p^{2} T^{10} + 23488 p^{4} T^{12} + 217 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 5 T + 58 T^{2} - 179 T^{3} + 3079 T^{4} - 179 p T^{5} + 58 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 + 53 T^{2} - 606 T^{4} - 149165 T^{6} - 8301346 T^{8} - 260309166 T^{10} + 2955616105 T^{12} + 815118488912 T^{14} + 45401270346297 T^{16} + 815118488912 p^{2} T^{18} + 2955616105 p^{4} T^{20} - 260309166 p^{6} T^{22} - 8301346 p^{8} T^{24} - 149165 p^{10} T^{26} - 606 p^{12} T^{28} + 53 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 232 T^{2} + 24618 T^{4} - 1815014 T^{6} + 124253255 T^{8} - 8284516923 T^{10} + 519890455663 T^{12} - 31837101584005 T^{14} + 1808174634489315 T^{16} - 31837101584005 p^{2} T^{18} + 519890455663 p^{4} T^{20} - 8284516923 p^{6} T^{22} + 124253255 p^{8} T^{24} - 1815014 p^{10} T^{26} + 24618 p^{12} T^{28} - 232 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 272 T^{2} + 33374 T^{4} - 3120238 T^{6} + 295626995 T^{8} - 24770742499 T^{10} + 1741894126659 T^{12} - 117285517486725 T^{14} + 7429142089346515 T^{16} - 117285517486725 p^{2} T^{18} + 1741894126659 p^{4} T^{20} - 24770742499 p^{6} T^{22} + 295626995 p^{8} T^{24} - 3120238 p^{10} T^{26} + 33374 p^{12} T^{28} - 272 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 14 T + 47 T^{2} - 154 T^{3} + 590 T^{4} + 23576 T^{5} + 12087 T^{6} - 1191834 T^{7} + 997843 T^{8} - 1191834 p T^{9} + 12087 p^{2} T^{10} + 23576 p^{3} T^{11} + 590 p^{4} T^{12} - 154 p^{5} T^{13} + 47 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 9 T - 125 T^{2} + 420 T^{3} + 14494 T^{4} + 8361 T^{5} - 1191428 T^{6} + 221142 T^{7} + 65070439 T^{8} + 221142 p T^{9} - 1191428 p^{2} T^{10} + 8361 p^{3} T^{11} + 14494 p^{4} T^{12} + 420 p^{5} T^{13} - 125 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 263 T^{2} + 33875 T^{4} + 2995944 T^{6} + 221839643 T^{8} + 2995944 p^{2} T^{10} + 33875 p^{4} T^{12} + 263 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 164 T^{2} - 210 T^{3} + 10879 T^{4} + 24885 T^{5} - 867851 T^{6} - 832335 T^{7} + 84012439 T^{8} - 832335 p T^{9} - 867851 p^{2} T^{10} + 24885 p^{3} T^{11} + 10879 p^{4} T^{12} - 210 p^{5} T^{13} - 164 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 10 T - 166 T^{2} - 742 T^{3} + 25697 T^{4} + 25865 T^{5} - 2591211 T^{6} - 2246829 T^{7} + 176237245 T^{8} - 2246829 p T^{9} - 2591211 p^{2} T^{10} + 25865 p^{3} T^{11} + 25697 p^{4} T^{12} - 742 p^{5} T^{13} - 166 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 67 T^{2} + 18572 T^{4} - 1111689 T^{6} + 164608199 T^{8} - 1111689 p^{2} T^{10} + 18572 p^{4} T^{12} - 67 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 412 T^{2} + 82722 T^{4} - 11456822 T^{6} + 1296977555 T^{8} - 127940738655 T^{10} + 11108997498859 T^{12} - 888294241775269 T^{14} + 74110599938321811 T^{16} - 888294241775269 p^{2} T^{18} + 11108997498859 p^{4} T^{20} - 127940738655 p^{6} T^{22} + 1296977555 p^{8} T^{24} - 11456822 p^{10} T^{26} + 82722 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 + 21 T + 380 T^{2} + 4851 T^{3} + 57837 T^{4} + 4851 p T^{5} + 380 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
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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
−2.21771515305325694609415247964, −2.17526712064444449399440035426, −2.14072803585353188156902097207, −2.04635997667860632584380826266, −2.01132677962845488735823373440, −1.81155922599837035495849039068, −1.78940804018938606819348666607, −1.78238360323112623639206820727, −1.77562229888699374098735664837, −1.66138324372166694391298773249, −1.44631182567147438648308416865, −1.27068460859540401287890688186, −1.22542791293841938259572076911, −1.13392656476277885499321753616, −1.13004932952135367765174253459, −1.11564810925173207104951502788, −1.00778777134383706611688229626, −0.980027760943562536926010481942, −0.820198794240767689006107913509, −0.793988881151274357379651167635, −0.48169145712646701790477119860, −0.39514033451078991229705312751, −0.33539225969038334544814429588, −0.29619063254712748247880875352, −0.07095593170571036653196875849, 0.07095593170571036653196875849, 0.29619063254712748247880875352, 0.33539225969038334544814429588, 0.39514033451078991229705312751, 0.48169145712646701790477119860, 0.793988881151274357379651167635, 0.820198794240767689006107913509, 0.980027760943562536926010481942, 1.00778777134383706611688229626, 1.11564810925173207104951502788, 1.13004932952135367765174253459, 1.13392656476277885499321753616, 1.22542791293841938259572076911, 1.27068460859540401287890688186, 1.44631182567147438648308416865, 1.66138324372166694391298773249, 1.77562229888699374098735664837, 1.78238360323112623639206820727, 1.78940804018938606819348666607, 1.81155922599837035495849039068, 2.01132677962845488735823373440, 2.04635997667860632584380826266, 2.14072803585353188156902097207, 2.17526712064444449399440035426, 2.21771515305325694609415247964
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.