Properties

Label 2-2475-2475.2056-c0-0-1
Degree $2$
Conductor $2475$
Sign $-0.475 + 0.879i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)5-s + (0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.499 − 0.866i)12-s + (−0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)20-s + (1.58 + 0.336i)23-s + (−0.499 + 0.866i)25-s + (0.309 − 0.951i)27-s + (−1.78 + 0.795i)31-s + (0.309 − 0.951i)33-s + (−0.809 − 0.587i)36-s + (−0.309 + 0.951i)37-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)5-s + (0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.499 − 0.866i)12-s + (−0.809 − 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)20-s + (1.58 + 0.336i)23-s + (−0.499 + 0.866i)25-s + (0.309 − 0.951i)27-s + (−1.78 + 0.795i)31-s + (0.309 − 0.951i)33-s + (−0.809 − 0.587i)36-s + (−0.309 + 0.951i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.475 + 0.879i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2056, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.500787408\)
\(L(\frac12)\) \(\approx\) \(1.500787408\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (0.104 + 0.994i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.104 - 0.994i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \)
29 \( 1 + (-0.669 - 0.743i)T^{2} \)
31 \( 1 + (1.78 - 0.795i)T + (0.669 - 0.743i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.104 - 0.994i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \)
53 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.895 - 0.994i)T + (-0.104 + 0.994i)T^{2} \)
61 \( 1 + (0.104 + 0.994i)T^{2} \)
67 \( 1 + (-1.66 + 0.743i)T + (0.669 - 0.743i)T^{2} \)
71 \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (0.978 + 0.207i)T^{2} \)
89 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (1.78 + 0.795i)T + (0.669 + 0.743i)T^{2} \)
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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

−8.795688757975369419475460900245, −8.427743379078500848758180047916, −7.26091325203933587151235270435, −6.74253358882940883334243726954, −5.65430513159977301787594127303, −4.94872911403720636872314112518, −3.96089362253465727851993115003, −3.16029873927837858612388533760, −1.74005489191758040511059021569, −0.968592042112737796904754500208, 2.06634662152155379846176729408, 2.89974706744363516036736577533, 3.74161533658578293843450524741, 4.17180796227022317162909903383, 5.22825898077336767136890959731, 6.79014803186273536191859444710, 7.15148281886275975386310181232, 7.81218639123244602422467905393, 8.619158226045608849501483923521, 9.233626523991961905630699585563

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