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Properties

Label 8-966e4-1.1-c3e4-0-1
Degree $8$
Conductor $870780120336$
Sign $1$
Analytic cond. $1.05529\times 10^{7}$
Root an. cond. $7.54955$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 12·3-s + 40·4-s − 5·5-s + 96·6-s − 28·7-s − 160·8-s + 90·9-s + 40·10-s − 41·11-s − 480·12-s + 23·13-s + 224·14-s + 60·15-s + 560·16-s + 18·17-s − 720·18-s + 15·19-s − 200·20-s + 336·21-s + 328·22-s + 92·23-s + 1.92e3·24-s − 128·25-s − 184·26-s − 540·27-s − 1.12e3·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s − 0.447·5-s + 6.53·6-s − 1.51·7-s − 7.07·8-s + 10/3·9-s + 1.26·10-s − 1.12·11-s − 11.5·12-s + 0.490·13-s + 4.27·14-s + 1.03·15-s + 35/4·16-s + 0.256·17-s − 9.42·18-s + 0.181·19-s − 2.23·20-s + 3.49·21-s + 3.17·22-s + 0.834·23-s + 16.3·24-s − 1.02·25-s − 1.38·26-s − 3.84·27-s − 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.05529\times 10^{7}\)
Root analytic conductor: \(7.54955\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
7$C_1$ \( ( 1 + p T )^{4} \)
23$C_1$ \( ( 1 - p T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + p T + 153 T^{2} + 37 p^{2} T^{3} + 23152 T^{4} + 37 p^{5} T^{5} + 153 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 41 T + 4131 T^{2} + 122684 T^{3} + 7235100 T^{4} + 122684 p^{3} T^{5} + 4131 p^{6} T^{6} + 41 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 23 T + 3537 T^{2} - 188273 T^{3} + 631420 p T^{4} - 188273 p^{3} T^{5} + 3537 p^{6} T^{6} - 23 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 18 T + 10963 T^{2} - 448 p^{2} T^{3} + 58927500 T^{4} - 448 p^{5} T^{5} + 10963 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 15 T + 27039 T^{2} - 306688 T^{3} + 14571052 p T^{4} - 306688 p^{3} T^{5} + 27039 p^{6} T^{6} - 15 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 46 T + 68301 T^{2} + 4146170 T^{3} + 2142947356 T^{4} + 4146170 p^{3} T^{5} + 68301 p^{6} T^{6} + 46 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 142 T + 71375 T^{2} - 3382160 T^{3} + 2105860112 T^{4} - 3382160 p^{3} T^{5} + 71375 p^{6} T^{6} - 142 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 142 T + 70125 T^{2} + 8566630 T^{3} + 5728192612 T^{4} + 8566630 p^{3} T^{5} + 70125 p^{6} T^{6} + 142 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 621 T + 403549 T^{2} + 138814946 T^{3} + 46400492174 T^{4} + 138814946 p^{3} T^{5} + 403549 p^{6} T^{6} + 621 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 185 T + 148161 T^{2} - 8597105 T^{3} + 9163416388 T^{4} - 8597105 p^{3} T^{5} + 148161 p^{6} T^{6} - 185 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 669 T + 255314 T^{2} - 63606245 T^{3} + 20219358106 T^{4} - 63606245 p^{3} T^{5} + 255314 p^{6} T^{6} - 669 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 422 T + 137022 T^{2} - 43069127 T^{3} + 17068551912 T^{4} - 43069127 p^{3} T^{5} + 137022 p^{6} T^{6} - 422 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 270 T + 580208 T^{2} - 81430245 T^{3} + 147472266354 T^{4} - 81430245 p^{3} T^{5} + 580208 p^{6} T^{6} - 270 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 272 T + 749802 T^{2} - 194861139 T^{3} + 237451298268 T^{4} - 194861139 p^{3} T^{5} + 749802 p^{6} T^{6} - 272 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + p T + 112055 T^{2} - 103369097 T^{3} - 30592203320 T^{4} - 103369097 p^{3} T^{5} + 112055 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 611 T + 673309 T^{2} + 48961993 T^{3} + 77099448980 T^{4} + 48961993 p^{3} T^{5} + 673309 p^{6} T^{6} - 611 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 236 T + 463173 T^{2} + 151020638 T^{3} + 337146652896 T^{4} + 151020638 p^{3} T^{5} + 463173 p^{6} T^{6} + 236 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 558 T + 1837711 T^{2} - 729539704 T^{3} + 1319686482496 T^{4} - 729539704 p^{3} T^{5} + 1837711 p^{6} T^{6} - 558 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 468 T + 1075673 T^{2} - 511126042 T^{3} + 973673977660 T^{4} - 511126042 p^{3} T^{5} + 1075673 p^{6} T^{6} - 468 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1519 T + 2148601 T^{2} + 2294362089 T^{3} + 2408098510620 T^{4} + 2294362089 p^{3} T^{5} + 2148601 p^{6} T^{6} + 1519 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 600 T + 2166211 T^{2} - 491792104 T^{3} + 2234813826424 T^{4} - 491792104 p^{3} T^{5} + 2166211 p^{6} T^{6} - 600 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.22899698409686857688417174838, −6.80124507705021708422323703943, −6.79876377346737294226680495217, −6.78610802612844182818151623672, −6.66187966324040824223316294657, −5.98611373666064363874901627442, −5.90521880418279776671311244070, −5.88083068199741334113374229986, −5.77679890768281982047449990847, −5.22207368499007073526681208158, −5.08404054993470113990299717380, −4.87227629895782597939501824455, −4.70246556094612531613182618492, −3.79698423103443998816551898522, −3.73793516851192387010246118006, −3.69135281863740859036419038045, −3.68340091906618008441523379597, −2.71749309113586332582390708313, −2.62261744342637951882252040247, −2.45318883725585326288850506069, −2.26662373835993323808859328217, −1.45326890216498943402841816006, −1.26043986684487275120994263187, −1.05236309864994985063539958768, −0.923642022706226285155562646246, 0, 0, 0, 0, 0.923642022706226285155562646246, 1.05236309864994985063539958768, 1.26043986684487275120994263187, 1.45326890216498943402841816006, 2.26662373835993323808859328217, 2.45318883725585326288850506069, 2.62261744342637951882252040247, 2.71749309113586332582390708313, 3.68340091906618008441523379597, 3.69135281863740859036419038045, 3.73793516851192387010246118006, 3.79698423103443998816551898522, 4.70246556094612531613182618492, 4.87227629895782597939501824455, 5.08404054993470113990299717380, 5.22207368499007073526681208158, 5.77679890768281982047449990847, 5.88083068199741334113374229986, 5.90521880418279776671311244070, 5.98611373666064363874901627442, 6.66187966324040824223316294657, 6.78610802612844182818151623672, 6.79876377346737294226680495217, 6.80124507705021708422323703943, 7.22899698409686857688417174838

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