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Properties

Label 2-950-95.44-c1-0-8
Degree $2$
Conductor $950$
Sign $-0.156 - 0.987i$
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 + (0.0291 − 0.0801i)3-s + (−0.173 + 0.984i)4-s + (0.0801 − 0.0291i)6-s + (−1.59 − 0.920i)7-s + (−0.866 + 0.500i)8-s + (2.29 + 1.92i)9-s + (−1.21 − 2.09i)11-s + (0.0738 + 0.0426i)12-s + (1.03 + 2.84i)13-s + (−0.319 − 1.81i)14-s + (−0.939 − 0.342i)16-s + (4.38 + 5.22i)17-s + 2.99i·18-s + (3.15 + 3.00i)19-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (0.0168 − 0.0462i)3-s + (−0.0868 + 0.492i)4-s + (0.0327 − 0.0119i)6-s + (−0.602 − 0.347i)7-s + (−0.306 + 0.176i)8-s + (0.764 + 0.641i)9-s + (−0.364 − 0.632i)11-s + (0.0213 + 0.0123i)12-s + (0.287 + 0.790i)13-s + (−0.0854 − 0.484i)14-s + (−0.234 − 0.0855i)16-s + (1.06 + 1.26i)17-s + 0.705i·18-s + (0.724 + 0.689i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.156 - 0.987i$
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: \((2,\ 950,\ (\ :1/2),\ -0.156 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19234 + 1.39627i\)
\(L(\frac12)\) \(\approx\) \(1.19234 + 1.39627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.15 - 3.00i)T \)
good3 \( 1 + (-0.0291 + 0.0801i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.59 + 0.920i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.21 + 2.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.03 - 2.84i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.38 - 5.22i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (7.90 + 1.39i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-7.77 - 6.52i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.41 - 4.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.202iT - 37T^{2} \)
41 \( 1 + (-5.41 - 1.97i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-2.07 + 0.365i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (6.32 - 7.53i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (6.17 + 1.08i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-4.62 + 3.87i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.741 + 4.20i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.00 - 2.39i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.94 + 11.0i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.67 + 7.33i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-9.87 - 3.59i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-11.0 - 6.37i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.21 + 0.442i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.27 + 6.28i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.27522937914252641930405370832, −9.495318868313617773381181434794, −8.210135682411889203694935273340, −7.85273305872058681181831098183, −6.70919486651678037599445777347, −6.08869131680451139220596765623, −5.07456943286814580890755896575, −4.01094249479198234770032263595, −3.24257997031279208060361832007, −1.58706621989355712370128953848, 0.78719966403556808801111432657, 2.44040180100399896994864278448, 3.35262474343613745779861709527, 4.36254800480516658089524530170, 5.39093808525089057761692676118, 6.20435477693718848892649641790, 7.24292843659618519236563073815, 8.066143002733737795699179176380, 9.577578256241436748003607528831, 9.657966802175240295021510154401

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