Properties

Label 2-950-25.16-c1-0-1
Degree $2$
Conductor $950$
Sign $-0.778 - 0.627i$
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  + (0.809 + 0.587i)2-s + (−0.432 − 1.33i)3-s + (0.309 + 0.951i)4-s + (−1.38 − 1.75i)5-s + (0.432 − 1.33i)6-s − 3.95·7-s + (−0.309 + 0.951i)8-s + (0.841 − 0.611i)9-s + (−0.0850 − 2.23i)10-s + (2.67 + 1.94i)11-s + (1.13 − 0.822i)12-s + (−4.62 + 3.35i)13-s + (−3.19 − 2.32i)14-s + (−1.74 + 2.60i)15-s + (−0.809 + 0.587i)16-s + (−0.364 + 1.12i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.249 − 0.768i)3-s + (0.154 + 0.475i)4-s + (−0.618 − 0.786i)5-s + (0.176 − 0.543i)6-s − 1.49·7-s + (−0.109 + 0.336i)8-s + (0.280 − 0.203i)9-s + (−0.0268 − 0.706i)10-s + (0.806 + 0.586i)11-s + (0.326 − 0.237i)12-s + (−1.28 + 0.931i)13-s + (−0.854 − 0.620i)14-s + (−0.449 + 0.671i)15-s + (−0.202 + 0.146i)16-s + (−0.0883 + 0.271i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.778 - 0.627i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.778 - 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111544 + 0.316381i\)
\(L(\frac12)\) \(\approx\) \(0.111544 + 0.316381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (1.38 + 1.75i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (0.432 + 1.33i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.95T + 7T^{2} \)
11 \( 1 + (-2.67 - 1.94i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.62 - 3.35i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.364 - 1.12i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (0.340 + 0.247i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.596 - 1.83i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.12 - 9.61i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.95 - 2.87i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.34 - 0.979i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 + (2.66 + 8.20i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.47 - 7.62i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.0592 - 0.0430i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.95 + 3.59i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.95 - 9.10i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.33 + 4.09i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.38 - 6.09i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.03 + 12.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.00 + 15.4i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (4.97 + 3.61i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.900 + 2.77i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.21513352513111172015978856782, −9.373080745196326721458092991717, −8.728228126752155138083974688607, −7.36800421169039509201334113924, −6.94562669432788272526504186005, −6.37128826985986734665622750871, −5.09703957128472772015667726188, −4.22029950000418659846210811062, −3.29156502069951637821573941786, −1.67664560072975427013094573132, 0.12802034191932137673326295988, 2.55186109476433260068102479061, 3.47720664067082608696414341941, 4.05454449873527206223556302833, 5.23566960688280494300718586062, 6.18394700865954840559613733478, 6.98707846077140516443752171768, 7.87184506617360327994257737897, 9.431890035703811011131475659406, 9.815687397467106939462592401707

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