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Properties

Label 8-147e4-1.1-c3e4-0-2
Degree $8$
Conductor $466948881$
Sign $1$
Analytic cond. $5658.92$
Root an. cond. $2.94504$
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  − 2·2-s + 6·3-s + 15·4-s − 20·5-s − 12·6-s − 50·8-s + 9·9-s + 40·10-s + 20·11-s + 90·12-s + 208·13-s − 120·15-s + 132·16-s − 116·17-s − 18·18-s − 192·19-s − 300·20-s − 40·22-s − 28·23-s − 300·24-s + 252·25-s − 416·26-s − 54·27-s + 592·29-s + 240·30-s + 104·31-s − 510·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 15/8·4-s − 1.78·5-s − 0.816·6-s − 2.20·8-s + 1/3·9-s + 1.26·10-s + 0.548·11-s + 2.16·12-s + 4.43·13-s − 2.06·15-s + 2.06·16-s − 1.65·17-s − 0.235·18-s − 2.31·19-s − 3.35·20-s − 0.387·22-s − 0.253·23-s − 2.55·24-s + 2.01·25-s − 3.13·26-s − 0.384·27-s + 3.79·29-s + 1.46·30-s + 0.602·31-s − 2.81·32-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(5658.92\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.902572796\)
\(L(\frac12)\) \(\approx\) \(3.902572796\)
\(L(\frac{5}{2})\) 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)$
bad3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7 \( 1 \)
good2$D_4\times C_2$ \( 1 + p T - 11 T^{2} - p T^{3} + 129 T^{4} - p^{4} T^{5} - 11 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
5$D_4\times C_2$ \( 1 + 4 p T + 148 T^{2} + 8 p T^{3} - 2121 T^{4} + 8 p^{4} T^{5} + 148 p^{6} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 20 T - 10 p^{2} T^{2} + 21040 T^{3} + 288139 T^{4} + 21040 p^{3} T^{5} - 10 p^{8} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 8 p T + 5848 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 116 T + 4316 T^{2} - 79576 T^{3} - 6707297 T^{4} - 79576 p^{3} T^{5} + 4316 p^{6} T^{6} + 116 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 192 T + 14898 T^{2} + 1583616 T^{3} + 182609099 T^{4} + 1583616 p^{3} T^{5} + 14898 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 28 T - 22178 T^{2} - 38416 T^{3} + 369678627 T^{4} - 38416 p^{3} T^{5} - 22178 p^{6} T^{6} + 28 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 296 T + 62994 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 104 T - 46470 T^{2} + 238784 T^{3} + 2071962659 T^{4} + 238784 p^{3} T^{5} - 46470 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 248 T - 50570 T^{2} - 2670464 T^{3} + 6879492955 T^{4} - 2670464 p^{3} T^{5} - 50570 p^{6} T^{6} - 248 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 20 T + 42020 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 720 T + 268614 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 96 T - 75734 T^{2} + 11778816 T^{3} - 4519545597 T^{4} + 11778816 p^{3} T^{5} - 75734 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 268 T + 15314 T^{2} - 64653392 T^{3} - 29663922677 T^{4} - 64653392 p^{3} T^{5} + 15314 p^{6} T^{6} + 268 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 616 T - 24014 T^{2} + 4489408 T^{3} + 42675213435 T^{4} + 4489408 p^{3} T^{5} - 24014 p^{6} T^{6} - 616 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 16 T - 453752 T^{2} + 736 T^{3} + 154544782567 T^{4} + 736 p^{3} T^{5} - 453752 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 144 T - 36822 T^{2} + 78331392 T^{3} - 93382080373 T^{4} + 78331392 p^{3} T^{5} - 36822 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 988 T + 852210 T^{2} - 988 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 104 T - 448320 T^{2} - 33165392 T^{3} + 55264032335 T^{4} - 33165392 p^{3} T^{5} - 448320 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 - 944 T - 206334 T^{2} - 105154048 T^{3} + 521988143075 T^{4} - 105154048 p^{3} T^{5} - 206334 p^{6} T^{6} - 944 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 1016 T + 1388838 T^{2} - 1016 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 388 T - 1067188 T^{2} + 74575928 T^{3} + 879761079727 T^{4} + 74575928 p^{3} T^{5} - 1067188 p^{6} T^{6} - 388 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 488 T + 1167280 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−8.701910731309028746738699624463, −8.691503985919520121280786033839, −8.588969546673608756166987780885, −8.248127960269625360064989132564, −8.224517840771322854769259508601, −8.093400022210695256861534801628, −7.55855865675264822404565869907, −6.87843094104987096644335900371, −6.59702807842329511629468500812, −6.53866062484784428824744584869, −6.53594560152330192005493333844, −6.23352535475125014326339958920, −6.04862655275858260729868642124, −5.11756047158219526999571850261, −4.69919174740475743883064768881, −4.44075063925451579994959583955, −3.76760873657305378819197924783, −3.60180789755293982454707128600, −3.60034832943202800300809158810, −3.02922431654482419398893196257, −2.56805111653701323577477677954, −2.15369609537345591673456908489, −1.68323557004784601096793565636, −0.988127304698986026293864756835, −0.53413835166077108109648753584, 0.53413835166077108109648753584, 0.988127304698986026293864756835, 1.68323557004784601096793565636, 2.15369609537345591673456908489, 2.56805111653701323577477677954, 3.02922431654482419398893196257, 3.60034832943202800300809158810, 3.60180789755293982454707128600, 3.76760873657305378819197924783, 4.44075063925451579994959583955, 4.69919174740475743883064768881, 5.11756047158219526999571850261, 6.04862655275858260729868642124, 6.23352535475125014326339958920, 6.53594560152330192005493333844, 6.53866062484784428824744584869, 6.59702807842329511629468500812, 6.87843094104987096644335900371, 7.55855865675264822404565869907, 8.093400022210695256861534801628, 8.224517840771322854769259508601, 8.248127960269625360064989132564, 8.588969546673608756166987780885, 8.691503985919520121280786033839, 8.701910731309028746738699624463

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