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Properties

Label 2-2475-2475.2254-c0-0-0
Degree $2$
Conductor $2475$
Sign $0.948 - 0.315i$
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  + (−0.104 − 0.994i)3-s + (−0.913 + 0.406i)4-s + (0.5 + 0.866i)5-s + (−0.978 + 0.207i)9-s + (0.978 − 0.207i)11-s + (0.499 + 0.866i)12-s + (0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.809 − 0.587i)20-s + (−0.873 + 0.786i)23-s + (−0.499 + 0.866i)25-s + (0.309 + 0.951i)27-s + (0.139 + 1.33i)31-s + (−0.309 − 0.951i)33-s + (0.809 − 0.587i)36-s + (1.64 − 0.535i)37-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)3-s + (−0.913 + 0.406i)4-s + (0.5 + 0.866i)5-s + (−0.978 + 0.207i)9-s + (0.978 − 0.207i)11-s + (0.499 + 0.866i)12-s + (0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (−0.809 − 0.587i)20-s + (−0.873 + 0.786i)23-s + (−0.499 + 0.866i)25-s + (0.309 + 0.951i)27-s + (0.139 + 1.33i)31-s + (−0.309 − 0.951i)33-s + (0.809 − 0.587i)36-s + (1.64 − 0.535i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.948 - 0.315i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.948 - 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001922585\)
\(L(\frac12)\) \(\approx\) \(1.001922585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
good2 \( 1 + (0.913 - 0.406i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.913 + 0.406i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.873 - 0.786i)T + (0.104 - 0.994i)T^{2} \)
29 \( 1 + (0.978 + 0.207i)T^{2} \)
31 \( 1 + (-0.139 - 1.33i)T + (-0.978 + 0.207i)T^{2} \)
37 \( 1 + (-1.64 + 0.535i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.913 - 0.406i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1.72 - 0.181i)T + (0.978 + 0.207i)T^{2} \)
53 \( 1 + (0.873 - 1.20i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-1.91 - 0.406i)T + (0.913 + 0.406i)T^{2} \)
61 \( 1 + (-0.913 + 0.406i)T^{2} \)
67 \( 1 + (1.97 - 0.207i)T + (0.978 - 0.207i)T^{2} \)
71 \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (0.669 + 0.743i)T^{2} \)
89 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (1.47 + 0.155i)T + (0.978 + 0.207i)T^{2} \)
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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

−9.069131148863050853390382671875, −8.384695166534064123183478630893, −7.50341212375953583046582336304, −6.98326655803548513397103937933, −6.02476947371123446579212162410, −5.54523806285105024616385527347, −4.24276179202808215630019357230, −3.38055460337315620688916891458, −2.44760806639702394957432294059, −1.21244203566417837792865106067, 0.834992779307046965694878795954, 2.29670674998840505579042732880, 3.84996962426151276846610310981, 4.31539472862584572286356981873, 4.99008339846206338410183181640, 5.88114692138764373357989021971, 6.32004849251717835809916928683, 7.920206265844331196227161606221, 8.571641436995710344086627827478, 9.282336835305981747057631084909

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