Properties

Degree $4$
Conductor $88510464$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s + 3·9-s + 4·11-s − 8·15-s − 12·17-s + 8·19-s − 4·23-s + 4·25-s − 4·27-s − 8·33-s − 8·37-s − 12·41-s + 12·45-s − 8·47-s + 24·51-s + 4·53-s + 16·55-s − 16·57-s + 8·61-s + 8·69-s − 4·71-s − 24·73-s − 8·75-s + 16·79-s + 5·81-s − 8·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s − 2.06·15-s − 2.91·17-s + 1.83·19-s − 0.834·23-s + 4/5·25-s − 0.769·27-s − 1.39·33-s − 1.31·37-s − 1.87·41-s + 1.78·45-s − 1.16·47-s + 3.36·51-s + 0.549·53-s + 2.15·55-s − 2.11·57-s + 1.02·61-s + 0.963·69-s − 0.474·71-s − 2.80·73-s − 0.923·75-s + 1.80·79-s + 5/9·81-s − 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 24 T + 272 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.13375207033334933103282385731, −7.03211411554798974718155845344, −6.68396337751472781930153468536, −6.63673566729484595563429461066, −6.13428839799472853123620851447, −5.83911055540854050457606803128, −5.44384045371663765909036901586, −5.35848736958174922667071932259, −4.78503278027433362046481685777, −4.58662223149930366718145262013, −4.00348074781870620256710503447, −3.84247250686774165316616474659, −3.24402490944947868600650685075, −2.67266292166849134808137184604, −2.25064078013074864030346091015, −1.83276791681258551697767480062, −1.37318924049428495399506006641, −1.30759791282426558021598316197, 0, 0, 1.30759791282426558021598316197, 1.37318924049428495399506006641, 1.83276791681258551697767480062, 2.25064078013074864030346091015, 2.67266292166849134808137184604, 3.24402490944947868600650685075, 3.84247250686774165316616474659, 4.00348074781870620256710503447, 4.58662223149930366718145262013, 4.78503278027433362046481685777, 5.35848736958174922667071932259, 5.44384045371663765909036901586, 5.83911055540854050457606803128, 6.13428839799472853123620851447, 6.63673566729484595563429461066, 6.68396337751472781930153468536, 7.03211411554798974718155845344, 7.13375207033334933103282385731

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