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Properties

Label 2-950-95.44-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.527 - 0.849i$
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  + (−0.642 − 0.766i)2-s + (−0.934 + 2.56i)3-s + (−0.173 + 0.984i)4-s + (2.56 − 0.934i)6-s + (1.85 + 1.06i)7-s + (0.866 − 0.500i)8-s + (−3.42 − 2.87i)9-s + (0.926 + 1.60i)11-s + (−2.36 − 1.36i)12-s + (1.88 + 5.18i)13-s + (−0.371 − 2.10i)14-s + (−0.939 − 0.342i)16-s + (2.37 + 2.82i)17-s + 4.46i·18-s + (−1.21 − 4.18i)19-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (−0.539 + 1.48i)3-s + (−0.0868 + 0.492i)4-s + (1.04 − 0.381i)6-s + (0.699 + 0.403i)7-s + (0.306 − 0.176i)8-s + (−1.14 − 0.957i)9-s + (0.279 + 0.483i)11-s + (−0.683 − 0.394i)12-s + (0.522 + 1.43i)13-s + (−0.0991 − 0.562i)14-s + (−0.234 − 0.0855i)16-s + (0.575 + 0.686i)17-s + 1.05i·18-s + (−0.278 − 0.960i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.527 - 0.849i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503692 + 0.906149i\)
\(L(\frac12)\) \(\approx\) \(0.503692 + 0.906149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (1.21 + 4.18i)T \)
good3 \( 1 + (0.934 - 2.56i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.85 - 1.06i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.926 - 1.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.88 - 5.18i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.37 - 2.82i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-2.89 - 0.511i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.36 - 4.50i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.12 + 3.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.49iT - 37T^{2} \)
41 \( 1 + (-1.47 - 0.536i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-2.29 + 0.404i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (7.20 - 8.59i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (6.32 + 1.11i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (10.8 - 9.07i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.0461 - 0.261i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (9.49 - 11.3i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.722 - 4.09i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-5.27 + 14.4i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (2.70 + 0.985i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (12.1 + 7.00i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.86 + 0.676i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-6.94 - 8.28i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.45073133968566398708850722477, −9.389706223394211341877195779413, −9.115069943989593234824243805426, −8.176993219327219572619041424727, −6.89449903479142694520461339793, −5.84858460379814899477321229988, −4.63249107417797519966144253056, −4.34096193871435531831439461928, −3.08155531209620219198410146130, −1.59186919910538234620711690794, 0.67863625419786163121414903514, 1.51423095018471742595703762549, 3.11626508140116904911021337693, 4.83188116584616031471213574818, 5.77629543427087630208225401319, 6.39819724326455508392213583656, 7.28697502948144346912989586681, 8.124268890241359817899666974912, 8.294879335902976530409799053617, 9.778659628883680252224485590323

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