Properties

Degree $4$
Conductor $2458624$
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  − 4·9-s − 4·11-s − 16·23-s − 10·25-s + 12·29-s − 4·37-s − 12·43-s + 12·53-s − 24·67-s + 8·71-s − 24·79-s + 7·81-s + 16·99-s − 8·107-s + 36·109-s − 24·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + ⋯
L(s)  = 1  − 4/3·9-s − 1.20·11-s − 3.33·23-s − 2·25-s + 2.22·29-s − 0.657·37-s − 1.82·43-s + 1.64·53-s − 2.93·67-s + 0.949·71-s − 2.70·79-s + 7/9·81-s + 1.60·99-s − 0.773·107-s + 3.44·109-s − 2.25·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1568} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 144 T^{2} + p^{2} T^{4} \)
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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

−9.038431681103424148294092035541, −8.796411830139782650515102904746, −8.332783511928667936434927338805, −8.121401070290854432438867330597, −7.66893629872196205128658087659, −7.52330076307777207232084780656, −6.62059559230917785167409095120, −6.36161183897516140287053002425, −5.82698469204577047203088655363, −5.71664717735909934406597119967, −5.15465608247286389502543564364, −4.70027223285448322345911122155, −3.91638252635517787482122209919, −3.90648630375322893589313542854, −2.98099328889607406564341366270, −2.65870680996730301452400797456, −2.12947067911120445145131286293, −1.52111127652180586649059427526, 0, 0, 1.52111127652180586649059427526, 2.12947067911120445145131286293, 2.65870680996730301452400797456, 2.98099328889607406564341366270, 3.90648630375322893589313542854, 3.91638252635517787482122209919, 4.70027223285448322345911122155, 5.15465608247286389502543564364, 5.71664717735909934406597119967, 5.82698469204577047203088655363, 6.36161183897516140287053002425, 6.62059559230917785167409095120, 7.52330076307777207232084780656, 7.66893629872196205128658087659, 8.121401070290854432438867330597, 8.332783511928667936434927338805, 8.796411830139782650515102904746, 9.038431681103424148294092035541

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