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Properties

Label 2-950-95.4-c1-0-21
Degree $2$
Conductor $950$
Sign $-0.471 + 0.881i$
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  + (0.342 − 0.939i)2-s + (−1.31 + 0.231i)3-s + (−0.766 − 0.642i)4-s + (−0.231 + 1.31i)6-s + (2.05 + 1.18i)7-s + (−0.866 + 0.500i)8-s + (−1.15 + 0.419i)9-s + (−1.33 − 2.31i)11-s + (1.15 + 0.665i)12-s + (0.955 + 0.168i)13-s + (1.81 − 1.52i)14-s + (0.173 + 0.984i)16-s + (1.93 − 5.30i)17-s + 1.22i·18-s + (−2.94 + 3.21i)19-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (−0.757 + 0.133i)3-s + (−0.383 − 0.321i)4-s + (−0.0944 + 0.535i)6-s + (0.776 + 0.448i)7-s + (−0.306 + 0.176i)8-s + (−0.384 + 0.139i)9-s + (−0.403 − 0.698i)11-s + (0.332 + 0.192i)12-s + (0.265 + 0.0467i)13-s + (0.485 − 0.407i)14-s + (0.0434 + 0.246i)16-s + (0.468 − 1.28i)17-s + 0.289i·18-s + (−0.674 + 0.737i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.471 + 0.881i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.471 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552375 - 0.921484i\)
\(L(\frac12)\) \(\approx\) \(0.552375 - 0.921484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 \)
19 \( 1 + (2.94 - 3.21i)T \)
good3 \( 1 + (1.31 - 0.231i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-2.05 - 1.18i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.955 - 0.168i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.93 + 5.30i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.49 + 6.55i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.0701 + 0.0255i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.00986 + 0.0170i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.17iT - 37T^{2} \)
41 \( 1 + (1.90 + 10.7i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (3.69 + 4.40i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (1.79 + 4.94i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.0572 + 0.0681i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-10.0 - 3.67i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (6.12 + 5.14i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-1.74 - 4.79i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (7.95 - 6.67i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (8.38 - 1.47i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (2.75 + 15.6i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.08 - 2.93i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.149 + 0.847i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.79 + 10.4i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.17155127778563843449836901014, −8.752214463578454620294082285421, −8.491931812635641646696290709228, −7.14129349168774181634586235232, −5.95332481770002905067716783036, −5.30510379657339847574918752300, −4.62386106905283270904855326257, −3.25513399439828301437705410912, −2.19932273569396752543062719794, −0.55134354585582508598976504513, 1.37454479228743708920726845417, 3.16511946844977595933624470383, 4.46038126388928691313451347520, 5.13333713396068732073581471560, 6.02789820612515474962001016500, 6.80715523071729202662670123655, 7.72628506596693667622616170289, 8.394421122695212054864876995983, 9.401732448243592973663157940832, 10.47014442832483771847468804141

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