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Properties

Label 2-2475-2475.263-c0-0-0
Degree $2$
Conductor $2475$
Sign $0.980 + 0.194i$
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  + (0.978 − 0.207i)3-s + (0.743 + 0.669i)4-s + (−0.866 − 0.5i)5-s + (0.913 − 0.406i)9-s + (−0.406 − 0.913i)11-s + (0.866 + 0.5i)12-s + (−0.951 − 0.309i)15-s + (0.104 + 0.994i)16-s + (−0.309 − 0.951i)20-s + (1.12 + 1.38i)23-s + (0.499 + 0.866i)25-s + (0.809 − 0.587i)27-s + (1.94 − 0.413i)31-s + (−0.587 − 0.809i)33-s + (0.951 + 0.309i)36-s + (0.302 − 1.90i)37-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)3-s + (0.743 + 0.669i)4-s + (−0.866 − 0.5i)5-s + (0.913 − 0.406i)9-s + (−0.406 − 0.913i)11-s + (0.866 + 0.5i)12-s + (−0.951 − 0.309i)15-s + (0.104 + 0.994i)16-s + (−0.309 − 0.951i)20-s + (1.12 + 1.38i)23-s + (0.499 + 0.866i)25-s + (0.809 − 0.587i)27-s + (1.94 − 0.413i)31-s + (−0.587 − 0.809i)33-s + (0.951 + 0.309i)36-s + (0.302 − 1.90i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.980 + 0.194i$
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: \((2,\ 2475,\ (\ :0),\ 0.980 + 0.194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.779716696\)
\(L(\frac12)\) \(\approx\) \(1.779716696\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + (-1.94 + 0.413i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (-0.302 + 1.90i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.669 + 0.743i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.281 + 0.434i)T + (-0.406 + 0.913i)T^{2} \)
53 \( 1 + (1.12 + 0.571i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (1.66 + 0.743i)T + (0.669 + 0.743i)T^{2} \)
61 \( 1 + (-0.669 + 0.743i)T^{2} \)
67 \( 1 + (0.913 - 1.40i)T + (-0.406 - 0.913i)T^{2} \)
71 \( 1 + (-0.773 + 0.251i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (1.30 - 0.846i)T + (0.406 - 0.913i)T^{2} \)
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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

−8.920047263961456696684574602224, −8.024816896610014132898722697408, −7.88297431501835362271066944484, −7.05879905661394656284093521166, −6.21414191598654297743027613479, −5.01471658635650105429628803803, −3.96393359259823394239498828425, −3.30840517773824879586292643255, −2.61741764124834686387301048193, −1.28098147619066197606956080340, 1.44099911947277582607402705474, 2.75349263353780078574048921300, 3.01086011151149243590207170861, 4.54086431533363286052952544500, 4.80569632349931672774970667798, 6.47975974396382518976129141827, 6.77066292640525229168398745951, 7.74950415685786870275026660067, 8.153836772140017788569630524275, 9.175196900343963118131156256764

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