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Properties

Label 2-2475-2475.2254-c0-0-1
Degree $2$
Conductor $2475$
Sign $-0.862 + 0.506i$
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  + (−0.669 − 0.743i)3-s + (−0.913 + 0.406i)4-s − 5-s + (−0.104 + 0.994i)9-s + (0.978 − 0.207i)11-s + (0.913 + 0.406i)12-s + (0.669 + 0.743i)15-s + (0.669 − 0.743i)16-s + (0.913 − 0.406i)20-s + (−0.873 + 0.786i)23-s + 25-s + (0.809 − 0.587i)27-s + (−0.204 − 1.94i)31-s + (−0.809 − 0.587i)33-s + (−0.309 − 0.951i)36-s + (−1.64 + 0.535i)37-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)3-s + (−0.913 + 0.406i)4-s − 5-s + (−0.104 + 0.994i)9-s + (0.978 − 0.207i)11-s + (0.913 + 0.406i)12-s + (0.669 + 0.743i)15-s + (0.669 − 0.743i)16-s + (0.913 − 0.406i)20-s + (−0.873 + 0.786i)23-s + 25-s + (0.809 − 0.587i)27-s + (−0.204 − 1.94i)31-s + (−0.809 − 0.587i)33-s + (−0.309 − 0.951i)36-s + (−1.64 + 0.535i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.862 + 0.506i$
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: \((2,\ 2475,\ (\ :0),\ -0.862 + 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2487310413\)
\(L(\frac12)\) \(\approx\) \(0.2487310413\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + T \)
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 \( 1 + (0.204 + 1.94i)T + (-0.978 + 0.207i)T^{2} \)
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 \( 1 + (0.244 - 0.336i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.30 + 0.278i)T + (0.913 + 0.406i)T^{2} \)
61 \( 1 + (-0.913 + 0.406i)T^{2} \)
67 \( 1 + (-0.809 + 0.0850i)T + (0.978 - 0.207i)T^{2} \)
71 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (0.669 + 0.743i)T^{2} \)
89 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.413 + 0.0434i)T + (0.978 + 0.207i)T^{2} \)
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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

−8.594184094318415029435279503974, −7.976351722129161009979796697843, −7.40083702036793748613741696937, −6.55560996545482505270559521745, −5.68038372183343744086024283443, −4.77295331608811778577521594469, −4.01461600214640607281515536712, −3.24284435542009453309828462813, −1.63938657734932327784713688819, −0.21254964168980173972838848706, 1.29340960265374727343533897888, 3.33278615520668977218756123607, 3.97165328988566226188785447832, 4.65706544891376728331321117699, 5.28055122322716806480862197693, 6.32444794277108948180246520362, 6.95165570140908597938438241599, 8.143869209162053737091237127751, 8.823872663771388670040459714515, 9.348444217009343383185797073426

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