Properties

Label 8-76e4-1.1-c5e4-0-0
Degree $8$
Conductor $33362176$
Sign $1$
Analytic cond. $22074.7$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·3-s − 110·5-s + 30·7-s + 3·9-s + 706·11-s + 788·13-s − 1.10e3·15-s + 240·17-s − 1.44e3·19-s + 300·21-s + 5.88e3·23-s + 5.68e3·25-s + 362·27-s + 5.24e3·29-s − 860·31-s + 7.06e3·33-s − 3.30e3·35-s − 2.07e4·37-s + 7.88e3·39-s − 1.02e4·41-s − 1.25e4·43-s − 330·45-s − 4.82e3·47-s − 4.38e4·49-s + 2.40e3·51-s − 7.64e4·53-s − 7.76e4·55-s + ⋯
L(s)  = 1  + 0.641·3-s − 1.96·5-s + 0.231·7-s + 1/81·9-s + 1.75·11-s + 1.29·13-s − 1.26·15-s + 0.201·17-s − 0.917·19-s + 0.148·21-s + 2.31·23-s + 1.81·25-s + 0.0955·27-s + 1.15·29-s − 0.160·31-s + 1.12·33-s − 0.455·35-s − 2.48·37-s + 0.829·39-s − 0.948·41-s − 1.03·43-s − 0.0242·45-s − 0.318·47-s − 2.60·49-s + 0.129·51-s − 3.74·53-s − 3.46·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(33362176\)    =    \(2^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(22074.7\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{76} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 33362176,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.653200859\)
\(L(\frac12)\) \(\approx\) \(1.653200859\)
\(L(\frac{7}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + p^{2} T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 10 T + 97 T^{2} - 434 p T^{3} + 5848 p^{2} T^{4} - 434 p^{6} T^{5} + 97 p^{10} T^{6} - 10 p^{15} T^{7} + p^{20} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 22 p T + 6413 T^{2} + 8878 p^{2} T^{3} + 5767376 T^{4} + 8878 p^{7} T^{5} + 6413 p^{10} T^{6} + 22 p^{16} T^{7} + p^{20} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 30 T + 44752 T^{2} + 98856 p T^{3} + 881393049 T^{4} + 98856 p^{6} T^{5} + 44752 p^{10} T^{6} - 30 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 706 T + 658673 T^{2} - 323249858 T^{3} + 161127831824 T^{4} - 323249858 p^{5} T^{5} + 658673 p^{10} T^{6} - 706 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 788 T + 1186231 T^{2} - 700956980 T^{3} + 620682637204 T^{4} - 700956980 p^{5} T^{5} + 1186231 p^{10} T^{6} - 788 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 240 T + 1613282 T^{2} - 761498208 T^{3} + 1482281900259 T^{4} - 761498208 p^{5} T^{5} + 1613282 p^{10} T^{6} - 240 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 5884 T + 21411695 T^{2} - 54686158844 T^{3} + 123442690709456 T^{4} - 54686158844 p^{5} T^{5} + 21411695 p^{10} T^{6} - 5884 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 5240 T + 60856211 T^{2} - 253837964428 T^{3} + 1617188633134184 T^{4} - 253837964428 p^{5} T^{5} + 60856211 p^{10} T^{6} - 5240 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 860 T + 91400044 T^{2} + 62709027884 T^{3} + 3723403118650534 T^{4} + 62709027884 p^{5} T^{5} + 91400044 p^{10} T^{6} + 860 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 20732 T + 227807476 T^{2} + 1980389778452 T^{3} + 17275967393509126 T^{4} + 1980389778452 p^{5} T^{5} + 227807476 p^{10} T^{6} + 20732 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 10204 T + 388738856 T^{2} + 3038996205812 T^{3} + 64399227505813550 T^{4} + 3038996205812 p^{5} T^{5} + 388738856 p^{10} T^{6} + 10204 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 12554 T + 479648677 T^{2} + 4138530607910 T^{3} + 95434697410657180 T^{4} + 4138530607910 p^{5} T^{5} + 479648677 p^{10} T^{6} + 12554 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 4826 T + 630393053 T^{2} + 4163406192922 T^{3} + 185783159690442980 T^{4} + 4163406192922 p^{5} T^{5} + 630393053 p^{10} T^{6} + 4826 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 76484 T + 3631277615 T^{2} + 116460309293524 T^{3} + 2762604272081552756 T^{4} + 116460309293524 p^{5} T^{5} + 3631277615 p^{10} T^{6} + 76484 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 23898 T + 1720660361 T^{2} - 24733920983166 T^{3} + 1308640952006846328 T^{4} - 24733920983166 p^{5} T^{5} + 1720660361 p^{10} T^{6} - 23898 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 32482 T + 2031335641 T^{2} + 31965315437422 T^{3} + 1619420039494417348 T^{4} + 31965315437422 p^{5} T^{5} + 2031335641 p^{10} T^{6} + 32482 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 5022 T + 984310405 T^{2} - 20898766206198 T^{3} - 478919880995889060 T^{4} - 20898766206198 p^{5} T^{5} + 984310405 p^{10} T^{6} - 5022 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 121300 T + 12237102908 T^{2} - 737604792352148 T^{3} + 38052926318924245718 T^{4} - 737604792352148 p^{5} T^{5} + 12237102908 p^{10} T^{6} - 121300 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 104700 T + 8117942698 T^{2} + 501529670831376 T^{3} + 26088351394045555539 T^{4} + 501529670831376 p^{5} T^{5} + 8117942698 p^{10} T^{6} + 104700 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 117128 T + 14278334536 T^{2} - 951853078254248 T^{3} + 65526252116476751662 T^{4} - 951853078254248 p^{5} T^{5} + 14278334536 p^{10} T^{6} - 117128 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 92832 T + 13268062568 T^{2} - 894150738137088 T^{3} + 71415811204234063134 T^{4} - 894150738137088 p^{5} T^{5} + 13268062568 p^{10} T^{6} - 92832 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 5988 T - 2589975052 T^{2} + 17516631078900 T^{3} + 49528889947005313350 T^{4} + 17516631078900 p^{5} T^{5} - 2589975052 p^{10} T^{6} - 5988 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 22972 T + 12158221672 T^{2} - 310772507916916 T^{3} + \)\(18\!\cdots\!82\)\( T^{4} - 310772507916916 p^{5} T^{5} + 12158221672 p^{10} T^{6} - 22972 p^{15} T^{7} + p^{20} T^{8} \)
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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

−9.862517792953791019186471355687, −9.050439576922314909830268289978, −9.039202755430444181666244516576, −8.824502654686296245969860278778, −8.596283517633568583178472943800, −8.121656609414142602846882040521, −7.972373649925004474354322312758, −7.77926596899161802417837212272, −7.20521179131213236927060556530, −6.84120663752561090423812078603, −6.58451447593414980970943941913, −6.35828208620470384198163370998, −6.20547181813379291535045415702, −5.06819908938554341504096715126, −4.93379466306303632770095227429, −4.85605074952776546825579491217, −4.19606431572491288110441971294, −3.67637079863556572563571874213, −3.45441998824108242666951653188, −3.34583163915862463514125538093, −2.87019025177418279132846485714, −1.74145827011482975363307579674, −1.56753379389982755416183322392, −0.983451502948084401135874345507, −0.27305029903724779261781600937, 0.27305029903724779261781600937, 0.983451502948084401135874345507, 1.56753379389982755416183322392, 1.74145827011482975363307579674, 2.87019025177418279132846485714, 3.34583163915862463514125538093, 3.45441998824108242666951653188, 3.67637079863556572563571874213, 4.19606431572491288110441971294, 4.85605074952776546825579491217, 4.93379466306303632770095227429, 5.06819908938554341504096715126, 6.20547181813379291535045415702, 6.35828208620470384198163370998, 6.58451447593414980970943941913, 6.84120663752561090423812078603, 7.20521179131213236927060556530, 7.77926596899161802417837212272, 7.972373649925004474354322312758, 8.121656609414142602846882040521, 8.596283517633568583178472943800, 8.824502654686296245969860278778, 9.039202755430444181666244516576, 9.050439576922314909830268289978, 9.862517792953791019186471355687

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