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Properties

Label 2-950-25.21-c1-0-40
Degree $2$
Conductor $950$
Sign $-0.981 - 0.191i$
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 + (−1.61 − 1.17i)3-s + (−0.809 − 0.587i)4-s + (1.68 − 1.47i)5-s + (−1.61 + 1.17i)6-s + 0.288·7-s + (−0.809 + 0.587i)8-s + (0.304 + 0.935i)9-s + (−0.884 − 2.05i)10-s + (1.73 − 5.33i)11-s + (0.616 + 1.89i)12-s + (0.550 + 1.69i)13-s + (0.0891 − 0.274i)14-s + (−4.44 + 0.411i)15-s + (0.309 + 0.951i)16-s + (−0.303 + 0.220i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.932 − 0.677i)3-s + (−0.404 − 0.293i)4-s + (0.751 − 0.659i)5-s + (−0.659 + 0.478i)6-s + 0.109·7-s + (−0.286 + 0.207i)8-s + (0.101 + 0.311i)9-s + (−0.279 − 0.649i)10-s + (0.523 − 1.60i)11-s + (0.178 + 0.547i)12-s + (0.152 + 0.470i)13-s + (0.0238 − 0.0733i)14-s + (−1.14 + 0.106i)15-s + (0.0772 + 0.237i)16-s + (−0.0735 + 0.0534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.981 - 0.191i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.981 - 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117428 + 1.21812i\)
\(L(\frac12)\) \(\approx\) \(0.117428 + 1.21812i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 1.47i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (1.61 + 1.17i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 0.288T + 7T^{2} \)
11 \( 1 + (-1.73 + 5.33i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.550 - 1.69i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.303 - 0.220i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-2.65 + 8.17i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.91 + 4.29i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.98 - 2.16i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.37 - 7.31i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.59 - 7.99i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.136T + 43T^{2} \)
47 \( 1 + (-8.51 - 6.18i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.65 + 4.10i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.09 + 12.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.58 - 4.88i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.99 + 1.45i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.92 + 2.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.93 - 12.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.15 - 2.29i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.05 - 5.12i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.08 + 9.50i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-12.6 - 9.20i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.579169730899518655637487158088, −8.932252695726875165311402219566, −8.107512891134479855154500825983, −6.55076533373550993589188660363, −6.15145958442441544194948765147, −5.31171934222381906342747715753, −4.33048110424152998828198414085, −2.97901742173010793253027559043, −1.55159411688707150285058873322, −0.62839290588670528024961849062, 1.91506440890267692148642984931, 3.53173117594724186869932971139, 4.55761719837018099256937960703, 5.51187959783684283154086117961, 5.88099763307260736977349962564, 7.18588825138263331154288858605, 7.45186658490584403841645541729, 9.175418470153865881485986718064, 9.572980928910125891326541004817, 10.49866855664365627500942724185

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