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Properties

Label 8-2310e4-1.1-c3e4-0-5
Degree $8$
Conductor $2.847\times 10^{13}$
Sign $1$
Analytic cond. $3.45073\times 10^{8}$
Root an. cond. $11.6745$
Motivic weight $3$
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  + 8·2-s + 12·3-s + 40·4-s − 20·5-s + 96·6-s − 28·7-s + 160·8-s + 90·9-s − 160·10-s − 44·11-s + 480·12-s − 224·14-s − 240·15-s + 560·16-s + 12·17-s + 720·18-s − 2·19-s − 800·20-s − 336·21-s − 352·22-s + 1.92e3·24-s + 250·25-s + 540·27-s − 1.12e3·28-s + 212·29-s − 1.92e3·30-s + 130·31-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s − 1.51·7-s + 7.07·8-s + 10/3·9-s − 5.05·10-s − 1.20·11-s + 11.5·12-s − 4.27·14-s − 4.13·15-s + 35/4·16-s + 0.171·17-s + 9.42·18-s − 0.0241·19-s − 8.94·20-s − 3.49·21-s − 3.41·22-s + 16.3·24-s + 2·25-s + 3.84·27-s − 7.55·28-s + 1.35·29-s − 11.6·30-s + 0.753·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3.45073\times 10^{8}\)
Root analytic conductor: \(11.6745\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(226.7452951\)
\(L(\frac12)\) \(\approx\) \(226.7452951\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
7$C_1$ \( ( 1 + p T )^{4} \)
11$C_1$ \( ( 1 + p T )^{4} \)
good13$C_2 \wr S_4$ \( 1 + 4714 T^{2} - 119852 T^{3} + 10122498 T^{4} - 119852 p^{3} T^{5} + 4714 p^{6} T^{6} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 12 T + 11997 T^{2} - 32426 T^{3} + 74297608 T^{4} - 32426 p^{3} T^{5} + 11997 p^{6} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2 T + 23981 T^{2} - 12814 T^{3} + 234710636 T^{4} - 12814 p^{3} T^{5} + 23981 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 19937 T^{2} + 2550 T^{3} + 389680508 T^{4} + 2550 p^{3} T^{5} + 19937 p^{6} T^{6} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 212 T + 81825 T^{2} - 14753816 T^{3} + 2800350364 T^{4} - 14753816 p^{3} T^{5} + 81825 p^{6} T^{6} - 212 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 130 T + 76478 T^{2} - 13739626 T^{3} + 2752171394 T^{4} - 13739626 p^{3} T^{5} + 76478 p^{6} T^{6} - 130 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 260 T + 168506 T^{2} - 26399168 T^{3} + 11123229410 T^{4} - 26399168 p^{3} T^{5} + 168506 p^{6} T^{6} - 260 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 106 T + 71714 T^{2} - 8952478 T^{3} + 10336385706 T^{4} - 8952478 p^{3} T^{5} + 71714 p^{6} T^{6} - 106 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 268 T + 254745 T^{2} - 54477396 T^{3} + 28871151028 T^{4} - 54477396 p^{3} T^{5} + 254745 p^{6} T^{6} - 268 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 566 T + 476614 T^{2} - 174464374 T^{3} + 77418471986 T^{4} - 174464374 p^{3} T^{5} + 476614 p^{6} T^{6} - 566 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 454 T + 563853 T^{2} - 166367586 T^{3} + 119666811052 T^{4} - 166367586 p^{3} T^{5} + 563853 p^{6} T^{6} - 454 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 844 T + 474449 T^{2} - 160930628 T^{3} + 59904632660 T^{4} - 160930628 p^{3} T^{5} + 474449 p^{6} T^{6} - 844 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1054 T + 1132949 T^{2} - 662259878 T^{3} + 396563754756 T^{4} - 662259878 p^{3} T^{5} + 1132949 p^{6} T^{6} - 1054 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 78 T + 912776 T^{2} - 62446838 T^{3} + 375444133502 T^{4} - 62446838 p^{3} T^{5} + 912776 p^{6} T^{6} - 78 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 822 T + 919118 T^{2} - 778689726 T^{3} + 439035997154 T^{4} - 778689726 p^{3} T^{5} + 919118 p^{6} T^{6} - 822 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1312 T + 1306352 T^{2} - 804094832 T^{3} + 521311873182 T^{4} - 804094832 p^{3} T^{5} + 1306352 p^{6} T^{6} - 1312 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 422 T + 677102 T^{2} - 106068046 T^{3} + 402901289634 T^{4} - 106068046 p^{3} T^{5} + 677102 p^{6} T^{6} - 422 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1902 T + 3263277 T^{2} - 3400933234 T^{3} + 3082829723308 T^{4} - 3400933234 p^{3} T^{5} + 3263277 p^{6} T^{6} - 1902 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1778 T + 2076105 T^{2} - 1962313460 T^{3} + 1609252305088 T^{4} - 1962313460 p^{3} T^{5} + 2076105 p^{6} T^{6} - 1778 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 12 p T + 3431777 T^{2} - 2927591488 T^{3} + 4567409858852 T^{4} - 2927591488 p^{3} T^{5} + 3431777 p^{6} T^{6} - 12 p^{10} T^{7} + p^{12} T^{8} \)
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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

−6.19151907721353277178063923672, −5.49488144926698516410948053039, −5.49421801305857791865528813854, −5.38139631200695803328379964444, −5.25618516157438748912736789237, −4.69121073622281024329086861982, −4.48859380396499880813911198809, −4.48626320998423892085665902213, −4.41796656112284138293000825682, −3.84047966027595272913741527858, −3.76121749765957826383482234637, −3.75626052857974532518641760243, −3.62727421963813936562405081384, −3.05090184744003923638126834136, −3.02014072499788790566366055039, −2.96567420424793375143808315007, −2.84858368019239912437091855833, −2.20230257633742455069759550469, −2.12191749667935461945403269283, −2.11986598337009040265820149391, −2.03304037165036233936850932443, −0.813372658239397852911612549738, −0.75450801471119932085116450134, −0.74734443971299146326634685634, −0.72084604637246011281616597765, 0.72084604637246011281616597765, 0.74734443971299146326634685634, 0.75450801471119932085116450134, 0.813372658239397852911612549738, 2.03304037165036233936850932443, 2.11986598337009040265820149391, 2.12191749667935461945403269283, 2.20230257633742455069759550469, 2.84858368019239912437091855833, 2.96567420424793375143808315007, 3.02014072499788790566366055039, 3.05090184744003923638126834136, 3.62727421963813936562405081384, 3.75626052857974532518641760243, 3.76121749765957826383482234637, 3.84047966027595272913741527858, 4.41796656112284138293000825682, 4.48626320998423892085665902213, 4.48859380396499880813911198809, 4.69121073622281024329086861982, 5.25618516157438748912736789237, 5.38139631200695803328379964444, 5.49421801305857791865528813854, 5.49488144926698516410948053039, 6.19151907721353277178063923672

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