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Properties

Label 2-950-95.4-c1-0-8
Degree $2$
Conductor $950$
Sign $0.505 - 0.862i$
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 + (−2.51 + 0.443i)3-s + (−0.766 − 0.642i)4-s + (−0.443 + 2.51i)6-s + (3.95 + 2.28i)7-s + (−0.866 + 0.500i)8-s + (3.30 − 1.20i)9-s + (−1.71 − 2.96i)11-s + (2.21 + 1.27i)12-s + (−0.803 − 0.141i)13-s + (3.49 − 2.93i)14-s + (0.173 + 0.984i)16-s + (−1.45 + 3.99i)17-s − 3.52i·18-s + (−4.31 + 0.637i)19-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (−1.45 + 0.256i)3-s + (−0.383 − 0.321i)4-s + (−0.181 + 1.02i)6-s + (1.49 + 0.862i)7-s + (−0.306 + 0.176i)8-s + (1.10 − 0.401i)9-s + (−0.516 − 0.894i)11-s + (0.638 + 0.368i)12-s + (−0.222 − 0.0392i)13-s + (0.934 − 0.784i)14-s + (0.0434 + 0.246i)16-s + (−0.352 + 0.968i)17-s − 0.830i·18-s + (−0.989 + 0.146i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.505 - 0.862i$
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: \((2,\ 950,\ (\ :1/2),\ 0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684466 + 0.392063i\)
\(L(\frac12)\) \(\approx\) \(0.684466 + 0.392063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.31 - 0.637i)T \)
good3 \( 1 + (2.51 - 0.443i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-3.95 - 2.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.71 + 2.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.803 + 0.141i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.45 - 3.99i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (3.10 - 3.70i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.25 + 0.820i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.45 + 5.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.82iT - 37T^{2} \)
41 \( 1 + (-2.12 - 12.0i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.83 - 3.38i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.54 - 6.98i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (7.25 - 8.64i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (5.29 + 1.92i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.36 - 6.17i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.92 + 5.29i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (11.7 - 9.87i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-9.16 + 1.61i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.376 - 2.13i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.416 - 0.240i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.474 + 2.68i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (0.682 - 1.87i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.60420926789415865679266675631, −9.637480738035536134243518628749, −8.442029378171713867437451930245, −7.923314170513569321151869083047, −6.06290915243504549935060881480, −5.93247213116391894406942928705, −4.82278217063845953750406238377, −4.30572280028148881840080958793, −2.60852702287153025955867498222, −1.33862315099982849290951441500, 0.44123468421553407051477539097, 2.06351817694540035630530280174, 4.22301415855204998432477728942, 4.83973023488235258724202268009, 5.37958253996289263004318383244, 6.61926460076625153793801817037, 7.13386747535284802877456080406, 7.909371619930371074519569226488, 8.866940870569716630377623129786, 10.34886731297011267653675947131

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