Properties

Label 4-1575e2-1.1-c1e2-0-5
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4·4-s + 2·7-s + 3·8-s + 2·11-s − 6·13-s + 6·14-s + 3·16-s − 2·17-s − 2·19-s + 6·22-s + 10·23-s − 18·26-s + 8·28-s + 4·29-s + 6·32-s − 6·34-s − 2·37-s − 6·38-s + 16·41-s − 12·43-s + 8·44-s + 30·46-s + 10·47-s + 3·49-s − 24·52-s − 8·53-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s + 0.755·7-s + 1.06·8-s + 0.603·11-s − 1.66·13-s + 1.60·14-s + 3/4·16-s − 0.485·17-s − 0.458·19-s + 1.27·22-s + 2.08·23-s − 3.53·26-s + 1.51·28-s + 0.742·29-s + 1.06·32-s − 1.02·34-s − 0.328·37-s − 0.973·38-s + 2.49·41-s − 1.82·43-s + 1.20·44-s + 4.42·46-s + 1.45·47-s + 3/7·49-s − 3.32·52-s − 1.09·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.942524841\)
\(L(\frac12)\) \(\approx\) \(7.942524841\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 12 T + 117 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 147 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 169 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−9.582215806854564896139889631601, −9.352575647383989737766304180376, −8.641134324747265075991133392320, −8.573498003149033809512235517840, −7.83726390652091623034848382756, −7.34756173834073922843620543774, −7.23770481946534508797662422676, −6.64940146965260254062409336537, −6.19053341951685855386750787312, −5.88447971383515596056400478327, −5.06741083821350428731945055235, −5.05550377763663725024136764167, −4.71504997871592873191306132131, −4.36871488897429737901495910206, −3.77458475511245606256615377174, −3.40938203382689387037834422273, −2.58658947243655109608344810028, −2.54164864007750832428835699876, −1.63322860491539759979092943201, −0.796389892251155548587007045794, 0.796389892251155548587007045794, 1.63322860491539759979092943201, 2.54164864007750832428835699876, 2.58658947243655109608344810028, 3.40938203382689387037834422273, 3.77458475511245606256615377174, 4.36871488897429737901495910206, 4.71504997871592873191306132131, 5.05550377763663725024136764167, 5.06741083821350428731945055235, 5.88447971383515596056400478327, 6.19053341951685855386750787312, 6.64940146965260254062409336537, 7.23770481946534508797662422676, 7.34756173834073922843620543774, 7.83726390652091623034848382756, 8.573498003149033809512235517840, 8.641134324747265075991133392320, 9.352575647383989737766304180376, 9.582215806854564896139889631601

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