Properties

Label 2-950-19.4-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.437 - 0.898i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.231 + 1.31i)3-s + (0.766 + 0.642i)4-s + (0.231 − 1.31i)6-s + (−2.18 + 3.78i)7-s + (−0.500 − 0.866i)8-s + (1.15 − 0.419i)9-s + (0.457 + 0.792i)11-s + (−0.665 + 1.15i)12-s + (0.700 − 3.97i)13-s + (3.34 − 2.81i)14-s + (0.173 + 0.984i)16-s + (5.73 + 2.08i)17-s − 1.22·18-s + (2.69 + 3.42i)19-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.133 + 0.757i)3-s + (0.383 + 0.321i)4-s + (0.0944 − 0.535i)6-s + (−0.826 + 1.43i)7-s + (−0.176 − 0.306i)8-s + (0.384 − 0.139i)9-s + (0.137 + 0.238i)11-s + (−0.192 + 0.332i)12-s + (0.194 − 1.10i)13-s + (0.895 − 0.751i)14-s + (0.0434 + 0.246i)16-s + (1.39 + 0.506i)17-s − 0.289·18-s + (0.618 + 0.785i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.437 - 0.898i$
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: \((2,\ 950,\ (\ :1/2),\ -0.437 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538250 + 0.860957i\)
\(L(\frac12)\) \(\approx\) \(0.538250 + 0.860957i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.69 - 3.42i)T \)
good3 \( 1 + (-0.231 - 1.31i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (2.18 - 3.78i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.457 - 0.792i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.700 + 3.97i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-5.73 - 2.08i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-0.676 - 0.567i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (8.44 - 3.07i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.11 - 5.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (-0.760 - 4.31i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-5.48 + 4.60i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.57 - 0.572i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-0.399 - 0.335i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-2.68 - 0.976i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.92 + 6.64i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (14.3 - 5.21i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.16 + 3.49i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.968 - 5.49i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.417 - 2.36i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.30 + 2.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.171 + 0.974i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-6.47 - 2.35i)T + (74.3 + 62.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.19643531683884593728248313097, −9.442764886369348923642124081320, −8.970474080764125611310387169937, −8.000295665366442947239162186694, −7.08246510171869974086463689705, −5.83018894154619963727177955838, −5.30323396281553757158116045264, −3.56126491117950300617519197412, −3.17844520969285043671612095221, −1.59906653071332837534710727567, 0.60942895527788271357511494028, 1.73635626787573009234595569413, 3.28158096198033879489873749269, 4.31599352370029610554988943719, 5.74838207035027429685129677386, 6.76450679172935694513283573671, 7.32656109867468023597118270908, 7.71416364563170133701549073434, 9.085147386435001378325656142746, 9.656438179255792415478285495371

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