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Properties

Label 2-2475-11.10-c0-0-2
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 11-s + 16-s − 2·31-s + 44-s + 49-s + 2·59-s + 64-s − 2·71-s − 2·89-s + ⋯
L(s)  = 1  + 4-s + 11-s + 16-s − 2·31-s + 44-s + 49-s + 2·59-s + 64-s − 2·71-s − 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2475} (901, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.621883571\)
\(L(\frac12)\) \(\approx\) \(1.621883571\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T^{2} \)
71 \( ( 1 + T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.088981671036900391281679885750, −8.365785984475252126636988029334, −7.31774909038249417865742996485, −6.97599818848143123231215191364, −6.03748792529617856815570508373, −5.42240454744561879636147355684, −4.15430717304387329961995238167, −3.39981170807814816649688143312, −2.31752175263114968694913267574, −1.36762491388352465914418801356, 1.36762491388352465914418801356, 2.31752175263114968694913267574, 3.39981170807814816649688143312, 4.15430717304387329961995238167, 5.42240454744561879636147355684, 6.03748792529617856815570508373, 6.97599818848143123231215191364, 7.31774909038249417865742996485, 8.365785984475252126636988029334, 9.088981671036900391281679885750

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