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Properties

Label 2-2475-2475.263-c0-0-1
Degree $2$
Conductor $2475$
Sign $0.975 - 0.221i$
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  + (0.994 + 0.104i)3-s + (0.743 + 0.669i)4-s − i·5-s + (0.978 + 0.207i)9-s + (0.406 + 0.913i)11-s + (0.669 + 0.743i)12-s + (0.104 − 0.994i)15-s + (0.104 + 0.994i)16-s + (0.669 − 0.743i)20-s + (−1.12 − 1.38i)23-s − 25-s + (0.951 + 0.309i)27-s + (−0.795 + 0.169i)31-s + (0.309 + 0.951i)33-s + (0.587 + 0.809i)36-s + (−0.0809 + 0.511i)37-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)3-s + (0.743 + 0.669i)4-s − i·5-s + (0.978 + 0.207i)9-s + (0.406 + 0.913i)11-s + (0.669 + 0.743i)12-s + (0.104 − 0.994i)15-s + (0.104 + 0.994i)16-s + (0.669 − 0.743i)20-s + (−1.12 − 1.38i)23-s − 25-s + (0.951 + 0.309i)27-s + (−0.795 + 0.169i)31-s + (0.309 + 0.951i)33-s + (0.587 + 0.809i)36-s + (−0.0809 + 0.511i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.048611839\)
\(L(\frac12)\) \(\approx\) \(2.048611839\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.994 - 0.104i)T \)
5 \( 1 + iT \)
11 \( 1 + (-0.406 - 0.913i)T \)
good2 \( 1 + (-0.743 - 0.669i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.743 - 0.669i)T^{2} \)
17 \( 1 + (-0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (1.12 + 1.38i)T + (-0.207 + 0.978i)T^{2} \)
29 \( 1 + (-0.913 - 0.406i)T^{2} \)
31 \( 1 + (0.795 - 0.169i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (0.0809 - 0.511i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.669 + 0.743i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (1.05 + 1.62i)T + (-0.406 + 0.913i)T^{2} \)
53 \( 1 + (-0.638 - 0.325i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.190 + 0.0850i)T + (0.669 + 0.743i)T^{2} \)
61 \( 1 + (-0.669 + 0.743i)T^{2} \)
67 \( 1 + (0.0570 - 0.0877i)T + (-0.406 - 0.913i)T^{2} \)
71 \( 1 + (-1.89 + 0.614i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.913 + 0.406i)T^{2} \)
83 \( 1 + (-0.994 - 0.104i)T^{2} \)
89 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-1.56 + 1.01i)T + (0.406 - 0.913i)T^{2} \)
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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

−8.936059625632659990468540739891, −8.371444303191145336433076760455, −7.78816429095354975431600238053, −7.01127443747766649199767178014, −6.26234121212487285697494696870, −4.95159131490042329548556396084, −4.20512082105818569895435576107, −3.49439733433892178422364578627, −2.30764480245171072702241152260, −1.66546404345292161421413273075, 1.50336553926806111336501737933, 2.37134396958130700402339424603, 3.28337166799745658716697206380, 3.89818195821432328494462643582, 5.37074443648673445222621200913, 6.19829639809395601737286089889, 6.78295366296963663406218057309, 7.59780607066039165077976430947, 8.120713599215195204530983441472, 9.290363522707871077278576036132

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