Properties

Label 8-354e4-1.1-c3e4-0-0
Degree $8$
Conductor $15704099856$
Sign $1$
Analytic cond. $190316.$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 12·3-s + 40·4-s + 22·5-s − 96·6-s + 13·7-s + 160·8-s + 90·9-s + 176·10-s + 24·11-s − 480·12-s + 20·13-s + 104·14-s − 264·15-s + 560·16-s + 91·17-s + 720·18-s + 141·19-s + 880·20-s − 156·21-s + 192·22-s + 13·23-s − 1.92e3·24-s + 131·25-s + 160·26-s − 540·27-s + 520·28-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s + 1.96·5-s − 6.53·6-s + 0.701·7-s + 7.07·8-s + 10/3·9-s + 5.56·10-s + 0.657·11-s − 11.5·12-s + 0.426·13-s + 1.98·14-s − 4.54·15-s + 35/4·16-s + 1.29·17-s + 9.42·18-s + 1.70·19-s + 9.83·20-s − 1.62·21-s + 1.86·22-s + 0.117·23-s − 16.3·24-s + 1.04·25-s + 1.20·26-s − 3.84·27-s + 3.50·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 59^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 59^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 59^{4}\)
Sign: $1$
Analytic conductor: \(190316.\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 59^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(57.04404180\)
\(L(\frac12)\) \(\approx\) \(57.04404180\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p T )^{4} \)
3$C_1$ \( ( 1 + p T )^{4} \)
59$C_1$ \( ( 1 - p T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 22 T + 353 T^{2} - 3566 T^{3} + 44256 T^{4} - 3566 p^{3} T^{5} + 353 p^{6} T^{6} - 22 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 13 T + 445 T^{2} - 7713 T^{3} + 292756 T^{4} - 7713 p^{3} T^{5} + 445 p^{6} T^{6} - 13 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 24 T + 4247 T^{2} - 72968 T^{3} + 7830664 T^{4} - 72968 p^{3} T^{5} + 4247 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 20 T + 5023 T^{2} - 200040 T^{3} + 12086572 T^{4} - 200040 p^{3} T^{5} + 5023 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 91 T + 13945 T^{2} - 703633 T^{3} + 76373716 T^{4} - 703633 p^{3} T^{5} + 13945 p^{6} T^{6} - 91 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 141 T + 22353 T^{2} - 2420173 T^{3} + 216954964 T^{4} - 2420173 p^{3} T^{5} + 22353 p^{6} T^{6} - 141 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 13 T + 32583 T^{2} - 860081 T^{3} + 500757040 T^{4} - 860081 p^{3} T^{5} + 32583 p^{6} T^{6} - 13 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 295 T + 122349 T^{2} - 22193701 T^{3} + 4780385256 T^{4} - 22193701 p^{3} T^{5} + 122349 p^{6} T^{6} - 295 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 311 T + 83889 T^{2} - 11594295 T^{3} + 2119986864 T^{4} - 11594295 p^{3} T^{5} + 83889 p^{6} T^{6} - 311 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 609 T + 213393 T^{2} - 64524355 T^{3} + 16928039968 T^{4} - 64524355 p^{3} T^{5} + 213393 p^{6} T^{6} - 609 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 677 T + 385201 T^{2} - 136176543 T^{3} + 42879690964 T^{4} - 136176543 p^{3} T^{5} + 385201 p^{6} T^{6} - 677 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 170 T + 233217 T^{2} - 40465226 T^{3} + 24503333332 T^{4} - 40465226 p^{3} T^{5} + 233217 p^{6} T^{6} - 170 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 17 T + 180841 T^{2} - 3574541 T^{3} + 17587740572 T^{4} - 3574541 p^{3} T^{5} + 180841 p^{6} T^{6} - 17 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 166 T + 345205 T^{2} - 24872398 T^{3} + 57567199144 T^{4} - 24872398 p^{3} T^{5} + 345205 p^{6} T^{6} - 166 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 651 T + 1051815 T^{2} - 456092713 T^{3} + 374818647004 T^{4} - 456092713 p^{3} T^{5} + 1051815 p^{6} T^{6} - 651 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 894 T + 1291385 T^{2} + 732845486 T^{3} + 581818799140 T^{4} + 732845486 p^{3} T^{5} + 1291385 p^{6} T^{6} + 894 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 298 T + 605489 T^{2} + 108737038 T^{3} + 123545557712 T^{4} + 108737038 p^{3} T^{5} + 605489 p^{6} T^{6} - 298 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 887 T + 1212003 T^{2} - 763795205 T^{3} + 638473439152 T^{4} - 763795205 p^{3} T^{5} + 1212003 p^{6} T^{6} - 887 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 784 T + 1331687 T^{2} + 1032029120 T^{3} + 839854247312 T^{4} + 1032029120 p^{3} T^{5} + 1331687 p^{6} T^{6} + 784 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 971 T + 2539367 T^{2} - 1665498531 T^{3} + 2246424166552 T^{4} - 1665498531 p^{3} T^{5} + 2539367 p^{6} T^{6} - 971 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1321 T + 2263745 T^{2} - 2253968011 T^{3} + 2269883328676 T^{4} - 2253968011 p^{3} T^{5} + 2263745 p^{6} T^{6} - 1321 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1922 T + 4082597 T^{2} + 4480899418 T^{3} + 5491899174668 T^{4} + 4480899418 p^{3} T^{5} + 4082597 p^{6} T^{6} + 1922 p^{9} T^{7} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.82995433875389415227432248594, −7.32020955697951291260191978365, −7.08743679401579342491536381058, −6.80770070532304743466576053444, −6.46034156152692299415959155117, −6.28770065835165885960493014785, −6.19290194846516381689590393492, −6.08409200164414383647167449692, −5.68171319410429827705041056598, −5.38268902547389256963234056544, −5.34391100365990801283579321314, −5.04551120434003619070125149114, −5.02194509148034312092344456750, −4.41528676509960890890291669222, −4.18278167560620773097132602589, −3.98459940382262151135628848689, −3.90339150951741775863672399576, −3.09038641828019862609143838496, −2.68553988297879590658569151272, −2.63646806207247064399166643160, −2.31767093845553973520191968782, −1.46972330159075926557417111432, −1.20258724379566111144012170101, −1.17559631612205724994606592085, −0.811021042959521902425660245932, 0.811021042959521902425660245932, 1.17559631612205724994606592085, 1.20258724379566111144012170101, 1.46972330159075926557417111432, 2.31767093845553973520191968782, 2.63646806207247064399166643160, 2.68553988297879590658569151272, 3.09038641828019862609143838496, 3.90339150951741775863672399576, 3.98459940382262151135628848689, 4.18278167560620773097132602589, 4.41528676509960890890291669222, 5.02194509148034312092344456750, 5.04551120434003619070125149114, 5.34391100365990801283579321314, 5.38268902547389256963234056544, 5.68171319410429827705041056598, 6.08409200164414383647167449692, 6.19290194846516381689590393492, 6.28770065835165885960493014785, 6.46034156152692299415959155117, 6.80770070532304743466576053444, 7.08743679401579342491536381058, 7.32020955697951291260191978365, 7.82995433875389415227432248594

Graph of the $Z$-function along the critical line