Properties

Label 4-1575e2-1.1-c1e2-0-4
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  + 2·2-s + 4-s + 2·7-s − 4·11-s + 4·13-s + 4·14-s + 16-s + 4·17-s − 8·22-s + 4·23-s + 8·26-s + 2·28-s − 16·29-s − 2·32-s + 8·34-s + 12·37-s + 4·41-s + 8·43-s − 4·44-s + 8·46-s − 8·47-s + 3·49-s + 4·52-s + 16·53-s − 32·58-s + 8·59-s + 12·61-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.755·7-s − 1.20·11-s + 1.10·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s − 1.70·22-s + 0.834·23-s + 1.56·26-s + 0.377·28-s − 2.97·29-s − 0.353·32-s + 1.37·34-s + 1.97·37-s + 0.624·41-s + 1.21·43-s − 0.603·44-s + 1.17·46-s − 1.16·47-s + 3/7·49-s + 0.554·52-s + 2.19·53-s − 4.20·58-s + 1.04·59-s + 1.53·61-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

\(L(1)\) \(\approx\) \(5.192782584\)
\(L(\frac12)\) \(\approx\) \(5.192782584\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_4$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−9.640639981803127140396868550522, −9.300043037037285924493162832455, −8.672139348674568336472164397788, −8.417711186473193927394013829548, −7.982400768327329245737691485838, −7.40817959548833171568646731378, −7.39917906707611315601409035028, −6.84874101280853142839110277633, −5.91750772154343333472196384105, −5.81909766951423116864094607071, −5.36203074373124717314864120749, −5.30865918421852678820916636976, −4.48312969709789838914568224774, −4.28388104909836250279335960684, −3.69680734059705540929043331236, −3.48643949906143382505202783646, −2.69623861641071732260051546833, −2.25668407264442344072386569946, −1.48755067666869266520752626945, −0.74640220026103708259968382766, 0.74640220026103708259968382766, 1.48755067666869266520752626945, 2.25668407264442344072386569946, 2.69623861641071732260051546833, 3.48643949906143382505202783646, 3.69680734059705540929043331236, 4.28388104909836250279335960684, 4.48312969709789838914568224774, 5.30865918421852678820916636976, 5.36203074373124717314864120749, 5.81909766951423116864094607071, 5.91750772154343333472196384105, 6.84874101280853142839110277633, 7.39917906707611315601409035028, 7.40817959548833171568646731378, 7.982400768327329245737691485838, 8.417711186473193927394013829548, 8.672139348674568336472164397788, 9.300043037037285924493162832455, 9.640639981803127140396868550522

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