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Properties

Label 2-950-95.3-c1-0-11
Degree $2$
Conductor $950$
Sign $-0.913 - 0.407i$
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  + (0.0871 + 0.996i)2-s + (−0.794 + 1.70i)3-s + (−0.984 + 0.173i)4-s + (−1.76 − 0.642i)6-s + (3.76 + 1.00i)7-s + (−0.258 − 0.965i)8-s + (−0.342 − 0.407i)9-s + (−1.56 + 2.71i)11-s + (0.486 − 1.81i)12-s + (1.46 − 0.684i)13-s + (−0.676 + 3.83i)14-s + (0.939 − 0.342i)16-s + (6.30 − 0.551i)17-s + (0.376 − 0.376i)18-s + (2.70 + 3.41i)19-s + ⋯
L(s)  = 1  + (0.0616 + 0.704i)2-s + (−0.458 + 0.983i)3-s + (−0.492 + 0.0868i)4-s + (−0.720 − 0.262i)6-s + (1.42 + 0.380i)7-s + (−0.0915 − 0.341i)8-s + (−0.114 − 0.135i)9-s + (−0.472 + 0.817i)11-s + (0.140 − 0.524i)12-s + (0.407 − 0.189i)13-s + (−0.180 + 1.02i)14-s + (0.234 − 0.0855i)16-s + (1.52 − 0.133i)17-s + (0.0886 − 0.0886i)18-s + (0.621 + 0.783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.913 - 0.407i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323959 + 1.52047i\)
\(L(\frac12)\) \(\approx\) \(0.323959 + 1.52047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 - 0.996i)T \)
5 \( 1 \)
19 \( 1 + (-2.70 - 3.41i)T \)
good3 \( 1 + (0.794 - 1.70i)T + (-1.92 - 2.29i)T^{2} \)
7 \( 1 + (-3.76 - 1.00i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.56 - 2.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.46 + 0.684i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (-6.30 + 0.551i)T + (16.7 - 2.95i)T^{2} \)
23 \( 1 + (-0.0168 + 0.0240i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (-0.496 + 0.416i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (6.55 - 3.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.88 - 4.88i)T + 37iT^{2} \)
41 \( 1 + (2.29 + 6.29i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (3.26 - 2.28i)T + (14.7 - 40.4i)T^{2} \)
47 \( 1 + (-1.16 + 13.3i)T + (-46.2 - 8.16i)T^{2} \)
53 \( 1 + (0.241 + 0.168i)T + (18.1 + 49.8i)T^{2} \)
59 \( 1 + (5.17 + 4.34i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.0829 + 0.470i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.83 - 0.685i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (5.21 + 0.918i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (5.79 + 2.70i)T + (46.9 + 55.9i)T^{2} \)
79 \( 1 + (11.2 - 4.10i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.42 - 12.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.79 + 1.38i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.733 - 8.38i)T + (-95.5 + 16.8i)T^{2} \)
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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

−10.20381544962205691892115206024, −9.752838773515806888507399834399, −8.577210883535348872320399878820, −7.88100483769074521476706583347, −7.18417739797723800085971656694, −5.60474616575541945344636782530, −5.32323048986700494405903140110, −4.52604236109299329635410041223, −3.47982560997018804218082602194, −1.65303273055706754594827718772, 0.866766277416394109754449272353, 1.64985578474027695264207275316, 3.08007143972488177929499235321, 4.29089856386737634864975867987, 5.37054573935156749408963508845, 6.04571087456291523107471051056, 7.46506016211274103930833931861, 7.78910451606496185562055401425, 8.797849222985303800803472091282, 9.827316481254904513700808200832

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