Properties

Label 4-10125-1.1-c1e2-0-0
Degree $4$
Conductor $10125$
Sign $1$
Analytic cond. $0.645578$
Root an. cond. $0.896369$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 5-s + 8·11-s + 5·16-s + 8·19-s + 3·20-s + 25-s + 4·29-s − 20·41-s − 24·44-s − 14·49-s − 8·55-s + 8·59-s − 4·61-s − 3·64-s + 16·71-s − 24·76-s − 5·80-s + 12·89-s − 8·95-s − 3·100-s − 12·101-s + 28·109-s − 12·116-s + 26·121-s − 125-s + 127-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.447·5-s + 2.41·11-s + 5/4·16-s + 1.83·19-s + 0.670·20-s + 1/5·25-s + 0.742·29-s − 3.12·41-s − 3.61·44-s − 2·49-s − 1.07·55-s + 1.04·59-s − 0.512·61-s − 3/8·64-s + 1.89·71-s − 2.75·76-s − 0.559·80-s + 1.27·89-s − 0.820·95-s − 0.299·100-s − 1.19·101-s + 2.68·109-s − 1.11·116-s + 2.36·121-s − 0.0894·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10125\)    =    \(3^{4} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(0.645578\)
Root analytic conductor: \(0.896369\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10125,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7596636170\)
\(L(\frac12)\) \(\approx\) \(0.7596636170\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−11.57318950916085697381955699128, −11.19897971912574422599178738873, −10.04517050900761317536685250745, −9.738851148469479131038939478590, −9.327820441539330420804608673051, −8.554588868470699894187777077269, −8.455181921164028207665051683361, −7.46261028773657489832215471574, −6.76955885158912340897756950945, −6.20681526904542088835730053039, −5.04860090093668311608273372623, −4.79492945983632365647604814265, −3.63412818236731287070311550143, −3.55714635244582915245852508954, −1.27025970316177507740293768624, 1.27025970316177507740293768624, 3.55714635244582915245852508954, 3.63412818236731287070311550143, 4.79492945983632365647604814265, 5.04860090093668311608273372623, 6.20681526904542088835730053039, 6.76955885158912340897756950945, 7.46261028773657489832215471574, 8.455181921164028207665051683361, 8.554588868470699894187777077269, 9.327820441539330420804608673051, 9.738851148469479131038939478590, 10.04517050900761317536685250745, 11.19897971912574422599178738873, 11.57318950916085697381955699128

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