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Properties

Label 2-950-25.21-c1-0-12
Degree $2$
Conductor $950$
Sign $-0.216 - 0.976i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (2.28 + 1.66i)3-s + (−0.809 − 0.587i)4-s + (0.312 + 2.21i)5-s + (2.28 − 1.66i)6-s − 5.10·7-s + (−0.809 + 0.587i)8-s + (1.53 + 4.73i)9-s + (2.20 + 0.386i)10-s + (−0.334 + 1.02i)11-s + (−0.872 − 2.68i)12-s + (0.0738 + 0.227i)13-s + (−1.57 + 4.85i)14-s + (−2.96 + 5.57i)15-s + (0.309 + 0.951i)16-s + (−3.12 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (1.31 + 0.958i)3-s + (−0.404 − 0.293i)4-s + (0.139 + 0.990i)5-s + (0.933 − 0.677i)6-s − 1.92·7-s + (−0.286 + 0.207i)8-s + (0.512 + 1.57i)9-s + (0.696 + 0.122i)10-s + (−0.100 + 0.310i)11-s + (−0.251 − 0.775i)12-s + (0.0204 + 0.0630i)13-s + (−0.421 + 1.29i)14-s + (−0.764 + 1.44i)15-s + (0.0772 + 0.237i)16-s + (−0.757 + 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.216 - 0.976i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.216 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07543 + 1.33978i\)
\(L(\frac12)\) \(\approx\) \(1.07543 + 1.33978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.312 - 2.21i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (-2.28 - 1.66i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 5.10T + 7T^{2} \)
11 \( 1 + (0.334 - 1.02i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.0738 - 0.227i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.12 - 2.26i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (0.561 - 1.72i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.48 - 3.25i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-6.99 + 5.08i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0153 + 0.0471i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.80 - 8.61i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.06T + 43T^{2} \)
47 \( 1 + (-4.02 - 2.92i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.679 + 0.494i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.92 - 8.98i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.32 + 7.14i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.80 + 4.94i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.65 + 4.10i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.0228 - 0.0703i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.45 + 3.96i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.97 + 1.43i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.92 - 5.91i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.09 - 4.43i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.10428091106597279490015999703, −9.679449533188916811018506198889, −8.996325434029999924568309614246, −8.002480021317477955121870459665, −6.77074730368744983036010261385, −6.07879840424955684158322499721, −4.48985481564789036707628544745, −3.64835538315095744723450937541, −3.01024233543979042316664118072, −2.34319122739516183414719099865, 0.60156706170086658505083138912, 2.43232223966681384998181583774, 3.28934647059330078009809678722, 4.34616501455670798365291596777, 5.72834570386374240631334613492, 6.65983835644319973720236500214, 7.09428516655649509150831407687, 8.364096193240347776181053316093, 8.662012935535890387406722546396, 9.464677508523446335820416346221

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