Properties

Label 2-950-19.11-c1-0-30
Degree $2$
Conductor $950$
Sign $-0.932 + 0.360i$
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 + (1.58 − 2.75i)3-s + (−0.499 − 0.866i)4-s + (−1.58 − 2.75i)6-s + 4.17·7-s − 0.999·8-s + (−3.54 − 6.14i)9-s − 3.90·11-s − 3.17·12-s + (0.239 + 0.414i)13-s + (2.08 − 3.61i)14-s + (−0.5 + 0.866i)16-s + (−0.245 + 0.424i)17-s − 7.09·18-s + (2.74 + 3.38i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.916 − 1.58i)3-s + (−0.249 − 0.433i)4-s + (−0.648 − 1.12i)6-s + 1.57·7-s − 0.353·8-s + (−1.18 − 2.04i)9-s − 1.17·11-s − 0.916·12-s + (0.0664 + 0.115i)13-s + (0.558 − 0.966i)14-s + (−0.125 + 0.216i)16-s + (−0.0594 + 0.102i)17-s − 1.67·18-s + (0.628 + 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.932 + 0.360i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487086 - 2.60832i\)
\(L(\frac12)\) \(\approx\) \(0.487086 - 2.60832i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.74 - 3.38i)T \)
good3 \( 1 + (-1.58 + 2.75i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4.17T + 7T^{2} \)
11 \( 1 + 3.90T + 11T^{2} \)
13 \( 1 + (-0.239 - 0.414i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.245 - 0.424i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.84 + 4.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.21 + 2.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.42T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 + (-0.254 + 0.441i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.91 - 5.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.485 + 0.841i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.77 - 4.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.82 + 6.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.65 - 8.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.809 - 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.937 + 1.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.35 - 4.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.19 - 5.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.96T + 83T^{2} \)
89 \( 1 + (-2.76 - 4.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.24 - 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.644779639437695552969510579316, −8.415978586546946312712158875782, −8.101724585022883471665449468421, −7.42757662650577397131711472252, −6.24802039084912624800348961745, −5.29933022031259130138476435105, −4.13378649904807268558391244010, −2.75033166233348761903272715832, −2.08135282434806657454811893382, −1.05954199163158067694578060148, 2.29993957966814936593166367025, 3.35199032371655453315089829298, 4.40498583447677662009698085908, 5.02814257503827655508400262211, 5.58933987562128882798752121445, 7.42801731976585291698899399931, 8.015005902791329264360018236450, 8.607158021674672231712908505111, 9.487352448308091121736321402435, 10.28305193633703135743207000755

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