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Properties

Label 2-2475-2475.2408-c0-0-0
Degree $2$
Conductor $2475$
Sign $0.942 - 0.335i$
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  + (−0.207 + 0.978i)3-s + (0.994 − 0.104i)4-s − i·5-s + (−0.913 − 0.406i)9-s + (0.743 + 0.669i)11-s + (−0.104 + 0.994i)12-s + (0.978 + 0.207i)15-s + (0.978 − 0.207i)16-s + (−0.104 − 0.994i)20-s + (−0.262 + 0.170i)23-s − 25-s + (0.587 − 0.809i)27-s + (1.35 − 0.604i)31-s + (−0.809 + 0.587i)33-s + (−0.951 − 0.309i)36-s + (0.877 + 1.72i)37-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)3-s + (0.994 − 0.104i)4-s − i·5-s + (−0.913 − 0.406i)9-s + (0.743 + 0.669i)11-s + (−0.104 + 0.994i)12-s + (0.978 + 0.207i)15-s + (0.978 − 0.207i)16-s + (−0.104 − 0.994i)20-s + (−0.262 + 0.170i)23-s − 25-s + (0.587 − 0.809i)27-s + (1.35 − 0.604i)31-s + (−0.809 + 0.587i)33-s + (−0.951 − 0.309i)36-s + (0.877 + 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.942 - 0.335i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.942 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.466672940\)
\(L(\frac12)\) \(\approx\) \(1.466672940\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.207 - 0.978i)T \)
5 \( 1 + iT \)
11 \( 1 + (-0.743 - 0.669i)T \)
good2 \( 1 + (-0.994 + 0.104i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.994 + 0.104i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.262 - 0.170i)T + (0.406 - 0.913i)T^{2} \)
29 \( 1 + (-0.669 - 0.743i)T^{2} \)
31 \( 1 + (-1.35 + 0.604i)T + (0.669 - 0.743i)T^{2} \)
37 \( 1 + (-0.877 - 1.72i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.104 + 0.994i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.185 + 0.483i)T + (-0.743 + 0.669i)T^{2} \)
53 \( 1 + (-0.0163 + 0.103i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (1.30 + 1.45i)T + (-0.104 + 0.994i)T^{2} \)
61 \( 1 + (0.104 + 0.994i)T^{2} \)
67 \( 1 + (-0.557 + 1.45i)T + (-0.743 - 0.669i)T^{2} \)
71 \( 1 + (0.244 + 0.336i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (0.669 + 0.743i)T^{2} \)
83 \( 1 + (0.207 - 0.978i)T^{2} \)
89 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (1.86 - 0.715i)T + (0.743 - 0.669i)T^{2} \)
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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

−9.495504907509139909445495336667, −8.368463971353619387743967218276, −7.81437369496318854494022110942, −6.59179142793454396374027674607, −6.09947028355629248928662140691, −5.11577847948480199996825291846, −4.46977400260974749566972235423, −3.59464646655163204550072345366, −2.47137525973764463924596153040, −1.25553489032873313605906640156, 1.26910796051327645042423040224, 2.41395978623673588585879605157, 2.99676426271166075859646040454, 4.05642006411195523306706362122, 5.67714582034048155404360131067, 6.12950292428650954412115519997, 6.80895106499620891419949616518, 7.35120944364382983273334106552, 8.063060077644376241197406924261, 8.874108759616000950530893733302

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