Properties

Degree $4$
Conductor $88510464$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

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: \(0\)
Selberg data: \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.494349653\)
\(L(\frac12)\) \(\approx\) \(4.494349653\)
\(L(\frac{3}{2})\) 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

−7.83219845016945941876324465716, −7.72275008542944268735737916750, −7.44808884543647556905944138475, −6.87284321770538830516674572001, −6.69769136073221782081088108468, −6.24760848126354811361698208045, −5.83749392489465073799054042500, −5.53735494953975402433748818696, −4.94298903220837828491623774324, −4.64463148681494043270224341480, −4.01135216538459630281032923746, −3.93086811399598343759646688153, −3.63222098131706243617607609724, −3.59402212957767371259867158690, −2.82219856392966073657043326428, −2.60236232474977328866187962993, −1.79749306378815904037374169628, −1.70652957926291254770572958719, −0.77103636407405851783647056237, −0.61369092890532234282127878369, 0.61369092890532234282127878369, 0.77103636407405851783647056237, 1.70652957926291254770572958719, 1.79749306378815904037374169628, 2.60236232474977328866187962993, 2.82219856392966073657043326428, 3.59402212957767371259867158690, 3.63222098131706243617607609724, 3.93086811399598343759646688153, 4.01135216538459630281032923746, 4.64463148681494043270224341480, 4.94298903220837828491623774324, 5.53735494953975402433748818696, 5.83749392489465073799054042500, 6.24760848126354811361698208045, 6.69769136073221782081088108468, 6.87284321770538830516674572001, 7.44808884543647556905944138475, 7.72275008542944268735737916750, 7.83219845016945941876324465716

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