Properties

Label 8-882e4-1.1-c2e4-0-3
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  + 4·4-s − 24·11-s + 12·16-s + 48·23-s + 28·25-s − 44·37-s + 28·43-s − 96·44-s + 240·53-s + 32·64-s − 220·67-s − 312·71-s + 20·79-s + 192·92-s + 112·100-s − 576·107-s + 580·109-s − 168·113-s − 124·121-s + 127-s + 131-s + 137-s + 139-s − 176·148-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 4-s − 2.18·11-s + 3/4·16-s + 2.08·23-s + 1.11·25-s − 1.18·37-s + 0.651·43-s − 2.18·44-s + 4.52·53-s + 1/2·64-s − 3.28·67-s − 4.39·71-s + 0.253·79-s + 2.08·92-s + 1.11·100-s − 5.38·107-s + 5.32·109-s − 1.48·113-s − 1.02·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.18·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1923482081\)
\(L(\frac12)\) \(\approx\) \(0.1923482081\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 28 T^{2} + 294 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{4} \)
13$D_4\times C_2$ \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 724 T^{2} + 279654 T^{4} - 724 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 58 p T^{2} + 550131 T^{4} - 58 p^{5} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 - 24 T + 554 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1678 T^{2} + 1875 p^{2} T^{4} - 1678 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 22 T + 2571 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 - 14 T + 3675 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 5884 T^{2} + 18275334 T^{4} - 5884 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 120 T + 8570 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 11476 T^{2} + 56933574 T^{4} - 11476 p^{4} T^{6} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 13252 T^{2} + 70932006 T^{4} - 13252 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 + 110 T + 8475 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 156 T + 178 p T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1390 T^{2} + 43340307 T^{4} - 1390 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 - 10 T + 3795 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 - 15410 T^{2} + p^{4} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \)
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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

−7.10553486110165627117568790482, −7.09203064107951781326828318087, −6.72187296788332888169177156418, −6.29202151900418024583195404286, −6.17578822789067103604565838499, −5.88999274050024501795214163125, −5.64150141299988136493282801900, −5.44478535288247550985165008458, −5.30423648262909056679696322236, −4.94129551597148886018286601703, −4.74999101737777855099276256939, −4.66196562839233124967753877586, −4.16533571095818743581149091183, −3.77778368017826621547247467668, −3.77247317859307510834462998771, −3.17229138258938035708769224771, −2.98138068712985321195794434446, −2.68718120906911471775970047035, −2.60748244694659129057416660886, −2.45748055188312596414284691610, −1.90602529453517399240091474974, −1.34402907799877051047905136245, −1.28872787804274720326488253844, −0.813772070787579504782818704325, −0.06373964186619437688015810750, 0.06373964186619437688015810750, 0.813772070787579504782818704325, 1.28872787804274720326488253844, 1.34402907799877051047905136245, 1.90602529453517399240091474974, 2.45748055188312596414284691610, 2.60748244694659129057416660886, 2.68718120906911471775970047035, 2.98138068712985321195794434446, 3.17229138258938035708769224771, 3.77247317859307510834462998771, 3.77778368017826621547247467668, 4.16533571095818743581149091183, 4.66196562839233124967753877586, 4.74999101737777855099276256939, 4.94129551597148886018286601703, 5.30423648262909056679696322236, 5.44478535288247550985165008458, 5.64150141299988136493282801900, 5.88999274050024501795214163125, 6.17578822789067103604565838499, 6.29202151900418024583195404286, 6.72187296788332888169177156418, 7.09203064107951781326828318087, 7.10553486110165627117568790482

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