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Properties

Label 2-950-19.4-c1-0-17
Degree $2$
Conductor $950$
Sign $0.992 + 0.120i$
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  + (0.939 + 0.342i)2-s + (−0.326 − 1.85i)3-s + (0.766 + 0.642i)4-s + (0.326 − 1.85i)6-s + (−1.53 + 2.65i)7-s + (0.500 + 0.866i)8-s + (−0.5 + 0.181i)9-s + (2.17 + 3.76i)11-s + (0.939 − 1.62i)12-s + (1 − 5.67i)13-s + (−2.34 + 1.96i)14-s + (0.173 + 0.984i)16-s + (1.93 + 0.705i)17-s − 0.532·18-s + (4.34 − 0.405i)19-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.188 − 1.06i)3-s + (0.383 + 0.321i)4-s + (0.133 − 0.755i)6-s + (−0.579 + 1.00i)7-s + (0.176 + 0.306i)8-s + (−0.166 + 0.0606i)9-s + (0.655 + 1.13i)11-s + (0.271 − 0.469i)12-s + (0.277 − 1.57i)13-s + (−0.627 + 0.526i)14-s + (0.0434 + 0.246i)16-s + (0.470 + 0.171i)17-s − 0.125·18-s + (0.995 − 0.0929i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32359 - 0.140057i\)
\(L(\frac12)\) \(\approx\) \(2.32359 - 0.140057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 \)
19 \( 1 + (-4.34 + 0.405i)T \)
good3 \( 1 + (0.326 + 1.85i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (1.53 - 2.65i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.17 - 3.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 5.67i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.93 - 0.705i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.94 - 4.98i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.87 - 1.41i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.22 + 7.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (0.0248 + 0.140i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (8.34 - 6.99i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-5.53 + 2.01i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.87 - 3.25i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (7.19 + 2.61i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (6.94 + 5.82i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-5.92 + 2.15i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.34 + 3.64i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.365 + 2.07i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.0641 + 0.363i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.376 - 0.652i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.308 + 1.75i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (10.0 + 3.65i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.866202729571461801624412975258, −9.261135619488192320624405895432, −7.897726861831033634152411976581, −7.46393407392679850380481099780, −6.49454217466062931533992955062, −5.82199135857894006094611016889, −5.02295879841093897242216757208, −3.56217065169551406960936215847, −2.60843368248389460779388954392, −1.27700485415371258243589419934, 1.17195710930025252630496032116, 3.12669326885103388004781938218, 3.84696970358394496291793081789, 4.52338261422179946755648728366, 5.50634210788937472319358077816, 6.60961982923952430778089888352, 7.15944470396064746254100234609, 8.694226508709390152531975255822, 9.439419143888641652775837701767, 10.17223508863754102464101141954

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