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Properties

Label 2-950-95.64-c1-0-26
Degree $2$
Conductor $950$
Sign $-0.390 + 0.920i$
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  + (−0.866 + 0.5i)2-s + (2.29 − 1.32i)3-s + (0.499 − 0.866i)4-s + (−1.32 + 2.29i)6-s − 3.64i·7-s + 0.999i·8-s + (2 − 3.46i)9-s − 4.64·11-s − 2.64i·12-s + (−1.73 − i)13-s + (1.82 + 3.15i)14-s + (−0.5 − 0.866i)16-s + 3.99i·18-s + (−1.67 − 4.02i)19-s + (−4.82 − 8.35i)21-s + (4.02 − 2.32i)22-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (1.32 − 0.763i)3-s + (0.249 − 0.433i)4-s + (−0.540 + 0.935i)6-s − 1.37i·7-s + 0.353i·8-s + (0.666 − 1.15i)9-s − 1.40·11-s − 0.763i·12-s + (−0.480 − 0.277i)13-s + (0.487 + 0.843i)14-s + (−0.125 − 0.216i)16-s + 0.942i·18-s + (−0.384 − 0.923i)19-s + (−1.05 − 1.82i)21-s + (0.857 − 0.495i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.390 + 0.920i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776873 - 1.17376i\)
\(L(\frac12)\) \(\approx\) \(0.776873 - 1.17376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
19 \( 1 + (1.67 + 4.02i)T \)
good3 \( 1 + (-2.29 + 1.32i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.64iT - 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.42 - 0.822i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.822 + 1.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
37 \( 1 + 0.354iT - 37T^{2} \)
41 \( 1 + (-0.145 - 0.252i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.77 + 5.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.77 - 2.17i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.8 - 6.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.96 + 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.468 - 0.811i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.559 - 0.322i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.35 - 2.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.47 + 0.854i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.93iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.21 + 1.85i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.588913649817786466971035114017, −8.773351837460151573476169381980, −7.962965374830625631534007628396, −7.33809099650202589458609919561, −7.03131706678621052490117152215, −5.56726335939891797000307870858, −4.31883873706271191512374954578, −3.03312080585753167749507277381, −2.13684367990245243669139803842, −0.64144370753660313461971927563, 2.19312011771416872208799565533, 2.65626233163272111340817305261, 3.70838220268578919356987380659, 4.91347996601414299478697813934, 5.89605514909735997733441142717, 7.41257975652683117677634009954, 8.112109662548329115559737924782, 8.819606419190625604849484619574, 9.294357940489130646222893636249, 10.15264098754884313674287286932

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