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Properties

Label 2-950-25.6-c1-0-39
Degree $2$
Conductor $950$
Sign $-0.799 + 0.600i$
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 + (1.98 − 1.44i)3-s + (−0.809 + 0.587i)4-s + (1.96 + 1.06i)5-s + (−1.98 − 1.44i)6-s − 3.97·7-s + (0.809 + 0.587i)8-s + (0.935 − 2.87i)9-s + (0.402 − 2.19i)10-s + (−1.61 − 4.96i)11-s + (−0.758 + 2.33i)12-s + (1.42 − 4.39i)13-s + (1.22 + 3.77i)14-s + (5.44 − 0.729i)15-s + (0.309 − 0.951i)16-s + (−1.21 − 0.880i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (1.14 − 0.833i)3-s + (−0.404 + 0.293i)4-s + (0.879 + 0.475i)5-s + (−0.810 − 0.589i)6-s − 1.50·7-s + (0.286 + 0.207i)8-s + (0.311 − 0.959i)9-s + (0.127 − 0.695i)10-s + (−0.486 − 1.49i)11-s + (−0.218 + 0.673i)12-s + (0.395 − 1.21i)13-s + (0.328 + 1.00i)14-s + (1.40 − 0.188i)15-s + (0.0772 − 0.237i)16-s + (−0.293 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.799 + 0.600i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.799 + 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552745 - 1.65784i\)
\(L(\frac12)\) \(\approx\) \(0.552745 - 1.65784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 - 1.06i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-1.98 + 1.44i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
11 \( 1 + (1.61 + 4.96i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.42 + 4.39i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.21 + 0.880i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (-0.570 - 1.75i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.42 + 3.93i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.35 + 3.89i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.33 + 10.2i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.117 + 0.361i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.07T + 43T^{2} \)
47 \( 1 + (3.14 - 2.28i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.12 - 4.45i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.60 - 4.95i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.86 - 5.74i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (10.1 + 7.36i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.06 + 2.22i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.68 - 14.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.5 + 8.37i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.42 - 5.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.888 - 2.73i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (6.37 - 4.63i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.487833312226520821763472755461, −9.086709729326178245398254842498, −8.137384393795673949694272575214, −7.38066183543473404463283511564, −6.25509743462989543774514485322, −5.67281763376174106221801215043, −3.61896903737880805690293365910, −2.93634846487503656489109033536, −2.41998659192132180401319227924, −0.74484368147573313696970785092, 1.93952971763766451367502333429, 3.09867131355146430438831680925, 4.27393754901562391358453970631, 4.96758899315907402604367513806, 6.38276908125962263033698544442, 6.79989966137360168739691930964, 8.093032380577336227574007059716, 9.027162553848941301232432974191, 9.355982109469680737836611038893, 9.964608832609194438630671727379

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