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Properties

Label 2-950-19.7-c1-0-7
Degree $2$
Conductor $950$
Sign $-0.431 - 0.901i$
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  + (0.5 + 0.866i)2-s + (−0.741 − 1.28i)3-s + (−0.499 + 0.866i)4-s + (0.741 − 1.28i)6-s − 0.482·7-s − 0.999·8-s + (0.400 − 0.693i)9-s − 4.43·11-s + 1.48·12-s + (−2.07 + 3.58i)13-s + (−0.241 − 0.418i)14-s + (−0.5 − 0.866i)16-s + (3.94 + 6.84i)17-s + 0.801·18-s + (4.31 − 0.590i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.428 − 0.741i)3-s + (−0.249 + 0.433i)4-s + (0.302 − 0.524i)6-s − 0.182·7-s − 0.353·8-s + (0.133 − 0.231i)9-s − 1.33·11-s + 0.428·12-s + (−0.574 + 0.994i)13-s + (−0.0645 − 0.111i)14-s + (−0.125 − 0.216i)16-s + (0.957 + 1.65i)17-s + 0.188·18-s + (0.990 − 0.135i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.431 - 0.901i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.431 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526528 + 0.835901i\)
\(L(\frac12)\) \(\approx\) \(0.526528 + 0.835901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-4.31 + 0.590i)T \)
good3 \( 1 + (0.741 + 1.28i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.482T + 7T^{2} \)
11 \( 1 + 4.43T + 11T^{2} \)
13 \( 1 + (2.07 - 3.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.94 - 6.84i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.82 - 4.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.91 - 3.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + 3.50T + 37T^{2} \)
41 \( 1 + (-4.44 - 7.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.318 - 0.551i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.13 + 3.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.59 - 2.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.812 + 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.735 - 1.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.59 - 11.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.41 + 2.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.43 - 5.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.576 - 0.997i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 + (4.37 - 7.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.476 - 0.825i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15328892412261477511239221792, −9.542383893167021773700990047366, −8.268810641194130464101559647698, −7.62413809424541398845772412973, −6.92057907279595052291285310656, −5.99793628805771921440183803103, −5.38100616254090652991364278710, −4.20219886036276935739782000853, −3.05872028856277631821557257919, −1.52149927251986706653108394257, 0.43635051499449360878986528826, 2.48328317857295522794370425816, 3.25736262727471465078150319555, 4.64246406751294526822753503812, 5.16298178067737065872467396393, 5.86739761564147567737272054206, 7.42230602093263606487128812251, 7.977736064698413461091821972167, 9.390255592253512280665887408314, 10.13119823060410553581722800937

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