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Properties

Label 2-950-25.6-c1-0-40
Degree $2$
Conductor $950$
Sign $-0.936 + 0.351i$
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 + (2.20 − 1.60i)3-s + (−0.809 + 0.587i)4-s + (−1.79 − 1.33i)5-s + (−2.20 − 1.60i)6-s + 2.97·7-s + (0.809 + 0.587i)8-s + (1.36 − 4.20i)9-s + (−0.710 + 2.12i)10-s + (−0.683 − 2.10i)11-s + (−0.841 + 2.58i)12-s + (−0.551 + 1.69i)13-s + (−0.918 − 2.82i)14-s + (−6.08 − 0.0548i)15-s + (0.309 − 0.951i)16-s + (−3.90 − 2.83i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (1.27 − 0.924i)3-s + (−0.404 + 0.293i)4-s + (−0.803 − 0.595i)5-s + (−0.899 − 0.653i)6-s + 1.12·7-s + (0.286 + 0.207i)8-s + (0.454 − 1.40i)9-s + (−0.224 + 0.670i)10-s + (−0.206 − 0.634i)11-s + (−0.242 + 0.747i)12-s + (−0.153 + 0.471i)13-s + (−0.245 − 0.755i)14-s + (−1.57 − 0.0141i)15-s + (0.0772 − 0.237i)16-s + (−0.947 − 0.688i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.936 + 0.351i$
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: \((2,\ 950,\ (\ :1/2),\ -0.936 + 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324322 - 1.78749i\)
\(L(\frac12)\) \(\approx\) \(0.324322 - 1.78749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 + 1.33i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-2.20 + 1.60i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 2.97T + 7T^{2} \)
11 \( 1 + (0.683 + 2.10i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.551 - 1.69i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.90 + 2.83i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (2.83 + 8.72i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.106 + 0.0775i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-8.08 - 5.87i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.66 + 8.20i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.339 + 1.04i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.21T + 43T^{2} \)
47 \( 1 + (-1.53 + 1.11i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (11.1 - 8.07i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.52 - 10.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.03 + 9.34i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-7.14 - 5.19i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-7.21 + 5.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.49 + 7.68i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.74 - 4.90i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.9 - 10.1i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.62 - 8.09i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (7.91 - 5.75i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.251641180923126298578077755564, −8.710089475331582254614103566840, −8.164401084183964782258279875239, −7.59684117587467168862498824368, −6.56441774658198243434001087503, −4.83797786746192807097043775398, −4.17841607813552119654500246539, −2.88009171058773744222960525905, −2.04837189229778309645121937405, −0.803712316636534666538814856917, 2.03870104127304813866238133745, 3.29645122822264890342504875697, 4.29485014752438089012653510910, 4.81178390911799493613854347515, 6.24400293919206223016185085865, 7.49254201165075933004823319262, 8.039411591391781885967826598160, 8.413106904724510524244149453187, 9.568076643873968884803659658819, 10.10362659316049848705158031283

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