Properties

Label 2-950-475.113-c1-0-36
Degree $2$
Conductor $950$
Sign $0.954 + 0.298i$
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  + (0.156 + 0.987i)2-s + (−0.311 − 0.158i)3-s + (−0.951 + 0.309i)4-s + (2.22 − 0.224i)5-s + (0.107 − 0.332i)6-s + (2.69 − 2.69i)7-s + (−0.453 − 0.891i)8-s + (−1.69 − 2.32i)9-s + (0.569 + 2.16i)10-s + (1.69 + 1.23i)11-s + (0.344 + 0.0546i)12-s + (−0.245 + 1.55i)13-s + (3.08 + 2.24i)14-s + (−0.727 − 0.282i)15-s + (0.809 − 0.587i)16-s + (−1.84 − 3.62i)17-s + ⋯
L(s)  = 1  + (0.110 + 0.698i)2-s + (−0.179 − 0.0915i)3-s + (−0.475 + 0.154i)4-s + (0.994 − 0.100i)5-s + (0.0440 − 0.135i)6-s + (1.02 − 1.02i)7-s + (−0.160 − 0.315i)8-s + (−0.563 − 0.776i)9-s + (0.180 + 0.683i)10-s + (0.512 + 0.372i)11-s + (0.0995 + 0.0157i)12-s + (−0.0681 + 0.430i)13-s + (0.825 + 0.599i)14-s + (−0.187 − 0.0730i)15-s + (0.202 − 0.146i)16-s + (−0.448 − 0.879i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.954 + 0.298i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81158 - 0.277144i\)
\(L(\frac12)\) \(\approx\) \(1.81158 - 0.277144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 - 0.987i)T \)
5 \( 1 + (-2.22 + 0.224i)T \)
19 \( 1 + (1.72 + 4.00i)T \)
good3 \( 1 + (0.311 + 0.158i)T + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (-2.69 + 2.69i)T - 7iT^{2} \)
11 \( 1 + (-1.69 - 1.23i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.245 - 1.55i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.84 + 3.62i)T + (-9.99 + 13.7i)T^{2} \)
23 \( 1 + (0.141 + 0.893i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (2.66 + 8.18i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.57 + 1.16i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.246 - 0.0390i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-5.39 - 7.42i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.615 + 0.615i)T + 43iT^{2} \)
47 \( 1 + (-1.81 - 0.922i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-4.06 - 2.07i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (3.76 - 2.73i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.90 - 6.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.38 - 1.72i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (-10.4 + 3.39i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.699 - 4.41i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-4.63 - 14.2i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-8.60 + 4.38i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (-6.46 - 4.69i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.43 - 8.69i)T + (-57.0 - 78.4i)T^{2} \)
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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

−9.674309906508952398679543902279, −9.241179021279555986562031334751, −8.298780103560596875011941338494, −7.22908691646885543470548255600, −6.64808250392416622418431852190, −5.76817155075174848300299770290, −4.76902187076723279785097681161, −4.06793280586561080141753648378, −2.38948527037928968288835475039, −0.894628733596025782386421597849, 1.68268809095688366957502905013, 2.30820452234174802628548125278, 3.63420526217890826507454760979, 5.05713670801108852318852967415, 5.50322157716865438109577359229, 6.30777363471334576696407944809, 7.83745841755831704012582604838, 8.708210287694750245990581928119, 9.134613990699718171745784587154, 10.41410739071727645549734617079

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