Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 8·19-s − 2·25-s − 4·27-s − 4·29-s + 12·37-s + 16·47-s − 12·53-s + 16·57-s − 24·59-s + 4·75-s + 5·81-s + 8·83-s + 8·87-s + 32·103-s − 20·109-s − 24·111-s + 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 32·141-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 1.83·19-s − 2/5·25-s − 0.769·27-s − 0.742·29-s + 1.97·37-s + 2.33·47-s − 1.64·53-s + 2.11·57-s − 3.12·59-s + 0.461·75-s + 5/9·81-s + 0.878·83-s + 0.857·87-s + 3.15·103-s − 1.91·109-s − 2.27·111-s + 3.38·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.69·141-s + 0.0819·149-s + 0.0813·151-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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 170 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−7.51770160447679466405554167716, −7.32183936039722784415471531742, −6.61757862339356182139985613531, −6.34935909758512881236318372356, −6.26258573794670946895669074931, −5.86716003977435541880860976679, −5.66467579258691980208546520947, −5.04008817464028213452904902084, −4.66121016363799056194743154946, −4.60721313110744683055976063586, −4.04028781612530651836348719963, −3.83666809803071275803343039271, −3.30462262386465517806987162958, −2.77489630792132571537210518484, −2.18230947785858181304749472356, −2.10214034970733727892519418987, −1.29007633253592092844614777481, −0.999600496970624257803597758027, 0, 0, 0.999600496970624257803597758027, 1.29007633253592092844614777481, 2.10214034970733727892519418987, 2.18230947785858181304749472356, 2.77489630792132571537210518484, 3.30462262386465517806987162958, 3.83666809803071275803343039271, 4.04028781612530651836348719963, 4.60721313110744683055976063586, 4.66121016363799056194743154946, 5.04008817464028213452904902084, 5.66467579258691980208546520947, 5.86716003977435541880860976679, 6.26258573794670946895669074931, 6.34935909758512881236318372356, 6.61757862339356182139985613531, 7.32183936039722784415471531742, 7.51770160447679466405554167716

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