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Properties

Label 8-7350e4-1.1-c1e4-0-3
Degree $8$
Conductor $2.918\times 10^{15}$
Sign $1$
Analytic cond. $1.18647\times 10^{7}$
Root an. cond. $7.66094$
Motivic weight $1$
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  + 4·2-s − 4·3-s + 10·4-s − 16·6-s + 20·8-s + 10·9-s − 40·12-s + 35·16-s − 4·17-s + 40·18-s − 12·19-s + 4·23-s − 80·24-s − 20·27-s − 8·29-s − 16·31-s + 56·32-s − 16·34-s + 100·36-s − 8·37-s − 48·38-s − 12·41-s + 4·43-s + 16·46-s − 12·47-s − 140·48-s + 16·51-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s − 6.53·6-s + 7.07·8-s + 10/3·9-s − 11.5·12-s + 35/4·16-s − 0.970·17-s + 9.42·18-s − 2.75·19-s + 0.834·23-s − 16.3·24-s − 3.84·27-s − 1.48·29-s − 2.87·31-s + 9.89·32-s − 2.74·34-s + 50/3·36-s − 1.31·37-s − 7.78·38-s − 1.87·41-s + 0.609·43-s + 2.35·46-s − 1.75·47-s − 20.2·48-s + 2.24·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.18647\times 10^{7}\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
7 \( 1 \)
good11$C_2 \wr C_2\wr C_2$ \( 1 + 26 T^{2} - 16 T^{3} + 362 T^{4} - 16 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T^{2} - 48 T^{3} + 322 T^{4} - 48 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 36 T^{2} + 196 T^{3} + 770 T^{4} + 196 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 88 T^{2} + 460 T^{3} + 2174 T^{4} + 460 p T^{5} + 88 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 56 T^{2} - 12 p T^{3} + 1646 T^{4} - 12 p^{2} T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 50 T^{2} + 360 T^{3} + 2786 T^{4} + 360 p T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 202 T^{2} + 1616 T^{3} + 10666 T^{4} + 1616 p T^{5} + 202 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 106 T^{2} + 768 T^{3} + 5306 T^{4} + 768 p T^{5} + 106 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 190 T^{2} + 1356 T^{3} + 11842 T^{4} + 1356 p T^{5} + 190 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 62 T^{2} - 148 T^{3} + 2370 T^{4} - 148 p T^{5} + 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 158 T^{2} + 1180 T^{3} + 9410 T^{4} + 1180 p T^{5} + 158 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 320 T^{2} - 3340 T^{3} + 28366 T^{4} - 3340 p T^{5} + 320 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 288 T^{2} + 2804 T^{3} + 24014 T^{4} + 2804 p T^{5} + 288 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 118 T^{2} + 684 T^{3} + 6306 T^{4} + 684 p T^{5} + 118 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 254 T^{2} + 788 T^{3} + 25090 T^{4} + 788 p T^{5} + 254 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 2788 p T^{5} + 248 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 206 T^{2} - 2092 T^{3} + 19170 T^{4} - 2092 p T^{5} + 206 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 140 T^{2} + 104 T^{3} + 6054 T^{4} + 104 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 244 T^{2} + 2056 T^{3} + 26854 T^{4} + 2056 p T^{5} + 244 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 44 T + 1014 T^{2} + 15436 T^{3} + 169698 T^{4} + 15436 p T^{5} + 1014 p^{2} T^{6} + 44 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 374 T^{2} + 3636 T^{3} + 43362 T^{4} + 3636 p T^{5} + 374 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.92967582607060374037944969119, −5.65184532349967667489511569328, −5.43655186203658199667844259874, −5.37525322800837279934571811160, −5.28776704716552498925970173015, −4.93538072176081317361978832660, −4.86447514616647970802260018501, −4.71478245740935848516468946931, −4.49306879765143448931231586555, −4.11927896146615383877093803438, −4.06958648429958007823782685821, −4.05652292991667983012299092060, −4.04924091514072851030845297654, −3.51970777289969006697885338378, −3.31676958416301699208828115401, −3.17494825253390800553898680160, −3.10604210413979497215114072767, −2.46008455231013063942692006750, −2.41291603631733658802479833019, −2.26636262797044787525677993064, −2.06745703963279135981140258921, −1.54788013190522080132128255705, −1.42424021545355517278086550576, −1.35523795673424963995086121854, −1.34206624696092089343684661190, 0, 0, 0, 0, 1.34206624696092089343684661190, 1.35523795673424963995086121854, 1.42424021545355517278086550576, 1.54788013190522080132128255705, 2.06745703963279135981140258921, 2.26636262797044787525677993064, 2.41291603631733658802479833019, 2.46008455231013063942692006750, 3.10604210413979497215114072767, 3.17494825253390800553898680160, 3.31676958416301699208828115401, 3.51970777289969006697885338378, 4.04924091514072851030845297654, 4.05652292991667983012299092060, 4.06958648429958007823782685821, 4.11927896146615383877093803438, 4.49306879765143448931231586555, 4.71478245740935848516468946931, 4.86447514616647970802260018501, 4.93538072176081317361978832660, 5.28776704716552498925970173015, 5.37525322800837279934571811160, 5.43655186203658199667844259874, 5.65184532349967667489511569328, 5.92967582607060374037944969119

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