Dirichlet series
| L(s) = 1 | − 8·3-s + 43·9-s + 16·19-s − 40·25-s − 124·27-s + 72·31-s − 40·37-s − 280·43-s + 56·49-s − 128·57-s − 56·61-s + 120·67-s − 208·73-s + 320·75-s + 204·79-s + 303·81-s − 576·93-s + 728·97-s + 168·103-s − 116·109-s + 320·111-s + 634·121-s + 127-s + 2.24e3·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 8/3·3-s + 43/9·9-s + 0.842·19-s − 8/5·25-s − 4.59·27-s + 2.32·31-s − 1.08·37-s − 6.51·43-s + 8/7·49-s − 2.24·57-s − 0.918·61-s + 1.79·67-s − 2.84·73-s + 4.26·75-s + 2.58·79-s + 3.74·81-s − 6.19·93-s + 7.50·97-s + 1.63·103-s − 1.06·109-s + 2.88·111-s + 5.23·121-s + 0.00787·127-s + 17.3·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.71799\times 10^{26}\) |
| Root analytic conductor: | \(6.76584\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(22.70782435\) |
| \(L(\frac12)\) | \(\approx\) | \(22.70782435\) |
| \(L(2)\) | 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 | \( 1 + 8 T + 7 p T^{2} - 52 T^{3} - 70 p^{2} T^{4} - 2116 T^{5} - 1381 T^{6} + 1936 p^{2} T^{7} + 1034 p^{4} T^{8} + 1936 p^{4} T^{9} - 1381 p^{4} T^{10} - 2116 p^{6} T^{11} - 70 p^{10} T^{12} - 52 p^{10} T^{13} + 7 p^{13} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 5 | \( ( 1 + p T^{2} )^{8} \) | |
| 7 | \( ( 1 - p T^{2} )^{8} \) | |
| good | 11 | \( 1 - 634 T^{2} + 247245 T^{4} - 65949338 T^{6} + 13712984198 T^{8} - 2284687822518 T^{10} + 326907972407363 T^{12} - 41733915025260470 T^{14} + 5125952848251765522 T^{16} - 41733915025260470 p^{4} T^{18} + 326907972407363 p^{8} T^{20} - 2284687822518 p^{12} T^{22} + 13712984198 p^{16} T^{24} - 65949338 p^{20} T^{26} + 247245 p^{24} T^{28} - 634 p^{28} T^{30} + p^{32} T^{32} \) |
| 13 | \( ( 1 + 745 T^{2} + 1452 T^{3} + 296782 T^{4} + 900988 T^{5} + 78625591 T^{6} + 264119992 T^{7} + 15370753026 T^{8} + 264119992 p^{2} T^{9} + 78625591 p^{4} T^{10} + 900988 p^{6} T^{11} + 296782 p^{8} T^{12} + 1452 p^{10} T^{13} + 745 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 1918 T^{2} + 1941581 T^{4} - 1392408966 T^{6} + 790147542694 T^{8} - 373036090829946 T^{10} + 150674753302219043 T^{12} - 52956031786998379138 T^{14} + \)\(16\!\cdots\!82\)\( T^{16} - 52956031786998379138 p^{4} T^{18} + 150674753302219043 p^{8} T^{20} - 373036090829946 p^{12} T^{22} + 790147542694 p^{16} T^{24} - 1392408966 p^{20} T^{26} + 1941581 p^{24} T^{28} - 1918 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - 8 T + 860 T^{2} - 16312 T^{3} + 704036 T^{4} - 10691240 T^{5} + 381169188 T^{6} - 5670939160 T^{7} + 156781324406 T^{8} - 5670939160 p^{2} T^{9} + 381169188 p^{4} T^{10} - 10691240 p^{6} T^{11} + 704036 p^{8} T^{12} - 16312 p^{10} T^{13} + 860 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 4480 T^{2} + 10302200 T^{4} - 16115718144 T^{6} + 19153465500700 T^{8} - 18300056706334464 T^{10} + 14511541318853927624 T^{12} - \)\(97\!\cdots\!40\)\( T^{14} + \)\(55\!\cdots\!30\)\( T^{16} - \)\(97\!\cdots\!40\)\( p^{4} T^{18} + 14511541318853927624 p^{8} T^{20} - 18300056706334464 p^{12} T^{22} + 19153465500700 p^{16} T^{24} - 16115718144 p^{20} T^{26} + 10302200 p^{24} T^{28} - 4480 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( 1 - 5378 T^{2} + 14503181 T^{4} - 26572841642 T^{6} + 38033992302502 T^{8} - 46226481721014678 T^{10} + 49944592354476620099 T^{12} - \)\(48\!\cdots\!42\)\( T^{14} + \)\(43\!\cdots\!90\)\( T^{16} - \)\(48\!\cdots\!42\)\( p^{4} T^{18} + 49944592354476620099 p^{8} T^{20} - 46226481721014678 p^{12} T^{22} + 38033992302502 p^{16} T^{24} - 26572841642 p^{20} T^{26} + 14503181 p^{24} T^{28} - 5378 p^{28} T^{30} + p^{32} T^{32} \) | |
| 31 | \( ( 1 - 36 T + 5588 T^{2} - 98172 T^{3} + 10424196 T^{4} + 32178444 T^{5} + 7586949868 T^{6} + 323190800532 T^{7} + 3540198882934 T^{8} + 323190800532 p^{2} T^{9} + 7586949868 p^{4} T^{10} + 32178444 p^{6} T^{11} + 10424196 p^{8} T^{12} - 98172 p^{10} T^{13} + 5588 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 20 T + 4860 T^{2} + 56172 T^{3} + 13001540 T^{4} + 138784868 T^{5} + 25923379716 T^{6} + 250888407260 T^{7} + 39435511353270 T^{8} + 250888407260 p^{2} T^{9} + 25923379716 p^{4} T^{10} + 138784868 p^{6} T^{11} + 13001540 p^{8} T^{12} + 56172 p^{10} T^{13} + 4860 p^{12} T^{14} + 20 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 - 17544 T^{2} + 152048344 T^{4} - 866284444952 T^{6} + 3637613347314524 T^{8} - 11949543415657926792 T^{10} + \)\(31\!\cdots\!80\)\( T^{12} - \)\(69\!\cdots\!84\)\( T^{14} + \)\(12\!\cdots\!02\)\( T^{16} - \)\(69\!\cdots\!84\)\( p^{4} T^{18} + \)\(31\!\cdots\!80\)\( p^{8} T^{20} - 11949543415657926792 p^{12} T^{22} + 3637613347314524 p^{16} T^{24} - 866284444952 p^{20} T^{26} + 152048344 p^{24} T^{28} - 17544 p^{28} T^{30} + p^{32} T^{32} \) | |
| 43 | \( ( 1 + 140 T + 17296 T^{2} + 1328244 T^{3} + 93523420 T^{4} + 4955438268 T^{5} + 256598382768 T^{6} + 10987637195140 T^{7} + 504821370222726 T^{8} + 10987637195140 p^{2} T^{9} + 256598382768 p^{4} T^{10} + 4955438268 p^{6} T^{11} + 93523420 p^{8} T^{12} + 1328244 p^{10} T^{13} + 17296 p^{12} T^{14} + 140 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 16494 T^{2} + 133830909 T^{4} - 714029015638 T^{6} + 2855140329276550 T^{8} - 9328199881448227946 T^{10} + \)\(26\!\cdots\!07\)\( T^{12} - \)\(67\!\cdots\!70\)\( T^{14} + \)\(15\!\cdots\!66\)\( T^{16} - \)\(67\!\cdots\!70\)\( p^{4} T^{18} + \)\(26\!\cdots\!07\)\( p^{8} T^{20} - 9328199881448227946 p^{12} T^{22} + 2855140329276550 p^{16} T^{24} - 714029015638 p^{20} T^{26} + 133830909 p^{24} T^{28} - 16494 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 - 26408 T^{2} + 323575832 T^{4} - 2478395195256 T^{6} + 13692147525400668 T^{8} - 60596495283388870952 T^{10} + \)\(43\!\cdots\!16\)\( p T^{12} - \)\(79\!\cdots\!44\)\( T^{14} + \)\(23\!\cdots\!78\)\( T^{16} - \)\(79\!\cdots\!44\)\( p^{4} T^{18} + \)\(43\!\cdots\!16\)\( p^{9} T^{20} - 60596495283388870952 p^{12} T^{22} + 13692147525400668 p^{16} T^{24} - 2478395195256 p^{20} T^{26} + 323575832 p^{24} T^{28} - 26408 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 - 14800 T^{2} + 138468568 T^{4} - 955392890736 T^{6} + 5479095487372636 T^{8} - 27019630879601870800 T^{10} + \)\(12\!\cdots\!12\)\( T^{12} - \)\(48\!\cdots\!04\)\( T^{14} + \)\(17\!\cdots\!62\)\( T^{16} - \)\(48\!\cdots\!04\)\( p^{4} T^{18} + \)\(12\!\cdots\!12\)\( p^{8} T^{20} - 27019630879601870800 p^{12} T^{22} + 5479095487372636 p^{16} T^{24} - 955392890736 p^{20} T^{26} + 138468568 p^{24} T^{28} - 14800 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 + 28 T + 12104 T^{2} + 279684 T^{3} + 87089292 T^{4} + 2698559212 T^{5} + 465312999544 T^{6} + 14439842194996 T^{7} + 1856098427255974 T^{8} + 14439842194996 p^{2} T^{9} + 465312999544 p^{4} T^{10} + 2698559212 p^{6} T^{11} + 87089292 p^{8} T^{12} + 279684 p^{10} T^{13} + 12104 p^{12} T^{14} + 28 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 60 T + 16740 T^{2} - 1235332 T^{3} + 154240932 T^{4} - 12326585708 T^{5} + 1033583040796 T^{6} - 78720265804692 T^{7} + 5329117931496566 T^{8} - 78720265804692 p^{2} T^{9} + 1033583040796 p^{4} T^{10} - 12326585708 p^{6} T^{11} + 154240932 p^{8} T^{12} - 1235332 p^{10} T^{13} + 16740 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( 1 - 17240 T^{2} + 219765336 T^{4} - 2041434479624 T^{6} + 16624899469572444 T^{8} - \)\(11\!\cdots\!96\)\( T^{10} + \)\(74\!\cdots\!40\)\( T^{12} - \)\(43\!\cdots\!72\)\( T^{14} + \)\(22\!\cdots\!98\)\( T^{16} - \)\(43\!\cdots\!72\)\( p^{4} T^{18} + \)\(74\!\cdots\!40\)\( p^{8} T^{20} - \)\(11\!\cdots\!96\)\( p^{12} T^{22} + 16624899469572444 p^{16} T^{24} - 2041434479624 p^{20} T^{26} + 219765336 p^{24} T^{28} - 17240 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 + 104 T + 24348 T^{2} + 2586552 T^{3} + 336158388 T^{4} + 30976570216 T^{5} + 3043939907492 T^{6} + 241340462677368 T^{7} + 19104376400195478 T^{8} + 241340462677368 p^{2} T^{9} + 3043939907492 p^{4} T^{10} + 30976570216 p^{6} T^{11} + 336158388 p^{8} T^{12} + 2586552 p^{10} T^{13} + 24348 p^{12} T^{14} + 104 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 102 T + 30141 T^{2} - 2523638 T^{3} + 463628630 T^{4} - 32866476410 T^{5} + 4655301343539 T^{6} - 284411378115370 T^{7} + 33720462322717682 T^{8} - 284411378115370 p^{2} T^{9} + 4655301343539 p^{4} T^{10} - 32866476410 p^{6} T^{11} + 463628630 p^{8} T^{12} - 2523638 p^{10} T^{13} + 30141 p^{12} T^{14} - 102 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( 1 - 42240 T^{2} + 975418872 T^{4} - 15454190107264 T^{6} + 185929829730979228 T^{8} - \)\(17\!\cdots\!28\)\( T^{10} + \)\(14\!\cdots\!28\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{14} + \)\(75\!\cdots\!98\)\( T^{16} - \)\(10\!\cdots\!48\)\( p^{4} T^{18} + \)\(14\!\cdots\!28\)\( p^{8} T^{20} - \)\(17\!\cdots\!28\)\( p^{12} T^{22} + 185929829730979228 p^{16} T^{24} - 15454190107264 p^{20} T^{26} + 975418872 p^{24} T^{28} - 42240 p^{28} T^{30} + p^{32} T^{32} \) | |
| 89 | \( 1 - 33032 T^{2} + 706967320 T^{4} - 12081015642520 T^{6} + 168525587092572636 T^{8} - \)\(20\!\cdots\!04\)\( T^{10} + \)\(21\!\cdots\!16\)\( T^{12} - \)\(20\!\cdots\!76\)\( T^{14} + \)\(16\!\cdots\!54\)\( T^{16} - \)\(20\!\cdots\!76\)\( p^{4} T^{18} + \)\(21\!\cdots\!16\)\( p^{8} T^{20} - \)\(20\!\cdots\!04\)\( p^{12} T^{22} + 168525587092572636 p^{16} T^{24} - 12081015642520 p^{20} T^{26} + 706967320 p^{24} T^{28} - 33032 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( ( 1 - 364 T + 112297 T^{2} - 23728036 T^{3} + 4389819350 T^{4} - 662305688748 T^{5} + 89325314408919 T^{6} - 10317093604637892 T^{7} + 1074975098720446610 T^{8} - 10317093604637892 p^{2} T^{9} + 89325314408919 p^{4} T^{10} - 662305688748 p^{6} T^{11} + 4389819350 p^{8} T^{12} - 23728036 p^{10} T^{13} + 112297 p^{12} T^{14} - 364 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−2.08765733290548440061663927595, −2.02681955973405054100022256224, −1.88925181775373141998780269370, −1.85301045019496603015013891526, −1.80297759816831795980184889099, −1.71631270300131136739635267168, −1.58880162307464775196608359750, −1.56798304843062293705536503136, −1.51277901449811757515638447422, −1.47631141495835733065772505870, −1.33770657762178648036350698761, −1.18750473835668477300688618579, −1.14650392732255885063870807376, −1.05091247850880121636218474420, −1.04706550196623868604821381806, −0.837193464589395541453778462792, −0.832967436496625601172640649330, −0.65914012324576684694194740724, −0.59614535642625525692385455627, −0.51729039684007278843377498334, −0.40400633574548717155817312631, −0.38520972994304185432129592198, −0.35733249889138005817515195399, −0.24608380285771759706605574354, −0.15058302156507178860671500075, 0.15058302156507178860671500075, 0.24608380285771759706605574354, 0.35733249889138005817515195399, 0.38520972994304185432129592198, 0.40400633574548717155817312631, 0.51729039684007278843377498334, 0.59614535642625525692385455627, 0.65914012324576684694194740724, 0.832967436496625601172640649330, 0.837193464589395541453778462792, 1.04706550196623868604821381806, 1.05091247850880121636218474420, 1.14650392732255885063870807376, 1.18750473835668477300688618579, 1.33770657762178648036350698761, 1.47631141495835733065772505870, 1.51277901449811757515638447422, 1.56798304843062293705536503136, 1.58880162307464775196608359750, 1.71631270300131136739635267168, 1.80297759816831795980184889099, 1.85301045019496603015013891526, 1.88925181775373141998780269370, 2.02681955973405054100022256224, 2.08765733290548440061663927595
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.