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Properties

Label 2-950-25.6-c1-0-25
Degree $2$
Conductor $950$
Sign $0.566 - 0.823i$
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 + (1.59 − 1.15i)3-s + (−0.809 + 0.587i)4-s + (2.11 + 0.730i)5-s + (1.59 + 1.15i)6-s − 1.09·7-s + (−0.809 − 0.587i)8-s + (0.270 − 0.833i)9-s + (−0.0412 + 2.23i)10-s + (1.19 + 3.66i)11-s + (−0.608 + 1.87i)12-s + (0.139 − 0.428i)13-s + (−0.337 − 1.03i)14-s + (4.21 − 1.28i)15-s + (0.309 − 0.951i)16-s + (4.87 + 3.53i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.919 − 0.668i)3-s + (−0.404 + 0.293i)4-s + (0.945 + 0.326i)5-s + (0.650 + 0.472i)6-s − 0.413·7-s + (−0.286 − 0.207i)8-s + (0.0903 − 0.277i)9-s + (−0.0130 + 0.706i)10-s + (0.359 + 1.10i)11-s + (−0.175 + 0.540i)12-s + (0.0386 − 0.118i)13-s + (−0.0902 − 0.277i)14-s + (1.08 − 0.331i)15-s + (0.0772 − 0.237i)16-s + (1.18 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.566 - 0.823i$
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: \((2,\ 950,\ (\ :1/2),\ 0.566 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33664 + 1.22901i\)
\(L(\frac12)\) \(\approx\) \(2.33664 + 1.22901i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 - 0.730i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-1.59 + 1.15i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 1.09T + 7T^{2} \)
11 \( 1 + (-1.19 - 3.66i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.139 + 0.428i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.87 - 3.53i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (1.49 + 4.60i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.0637 + 0.0462i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.37 + 0.996i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.786 - 2.42i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.59 + 4.92i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.64T + 43T^{2} \)
47 \( 1 + (-3.74 + 2.71i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.19 + 2.32i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.32 + 10.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.94 + 5.97i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (12.2 + 8.87i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.56 - 4.04i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.125 - 0.385i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.57 - 4.77i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.79 - 5.66i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.240 + 0.739i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.449 - 0.326i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.936412919736582776190511183602, −9.221937311346849968403309963372, −8.365123125700081055430805767910, −7.56295432763577986694657868941, −6.81451816043756748575472917606, −6.09636062623164676536799154494, −5.08768536491752237075430591263, −3.78889631498116285016571716968, −2.67936220399121538169560793773, −1.68529372023436369789711771328, 1.18627769896702281144602194403, 2.69429002206605712424383919902, 3.34180741265770495442919709851, 4.30961532395965479736030391931, 5.53247919023431799935043400065, 6.13154500076559189065813210996, 7.58909400959022207258020511713, 8.775683070607104997303791265098, 9.146545729724234066405684811366, 9.865589907106699183205391963031

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