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Properties

Label 2-950-19.4-c1-0-27
Degree $2$
Conductor $950$
Sign $-0.997 - 0.0667i$
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 + (−0.443 − 2.51i)3-s + (0.766 + 0.642i)4-s + (−0.443 + 2.51i)6-s + (2.28 − 3.95i)7-s + (−0.500 − 0.866i)8-s + (−3.30 + 1.20i)9-s + (−1.71 − 2.96i)11-s + (1.27 − 2.21i)12-s + (0.141 − 0.803i)13-s + (−3.49 + 2.93i)14-s + (0.173 + 0.984i)16-s + (3.99 + 1.45i)17-s + 3.52·18-s + (4.31 − 0.637i)19-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.256 − 1.45i)3-s + (0.383 + 0.321i)4-s + (−0.181 + 1.02i)6-s + (0.862 − 1.49i)7-s + (−0.176 − 0.306i)8-s + (−1.10 + 0.401i)9-s + (−0.516 − 0.894i)11-s + (0.368 − 0.638i)12-s + (0.0392 − 0.222i)13-s + (−0.934 + 0.784i)14-s + (0.0434 + 0.246i)16-s + (0.968 + 0.352i)17-s + 0.830·18-s + (0.989 − 0.146i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.997 - 0.0667i$
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: \((2,\ 950,\ (\ :1/2),\ -0.997 - 0.0667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0352112 + 1.05401i\)
\(L(\frac12)\) \(\approx\) \(0.0352112 + 1.05401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.31 + 0.637i)T \)
good3 \( 1 + (0.443 + 2.51i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-2.28 + 3.95i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.71 + 2.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.141 + 0.803i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.99 - 1.45i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (3.70 + 3.10i)T + (3.99 + 22.6i)T^{2} \)
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 \( 1 - 3.82T + 37T^{2} \)
41 \( 1 + (-2.12 - 12.0i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (3.38 - 2.83i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-6.98 + 2.54i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (8.64 + 7.25i)T + (9.20 + 52.1i)T^{2} \)
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 \( 1 + (5.29 - 1.92i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (11.7 - 9.87i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.61 - 9.16i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.376 + 2.13i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.240 - 0.416i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.474 - 2.68i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.87 - 0.682i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.821070602953258688963590192198, −8.273640283979998537847939312635, −7.921608383555948195465843879593, −7.37947550505518782586960136018, −6.43813487117778688891247857637, −5.51941536435253136569038879920, −4.08810819128309919790326472723, −2.78254454204746912468369394449, −1.37947219498082588661765638100, −0.69997520099654984691995053247, 1.88434521985263441511088817020, 3.14818952181270780492752766369, 4.53759567248177508532612672476, 5.35991000126961564702463558107, 5.76794512898398862573689831063, 7.35048146697612934803844990677, 8.109594676265755368073588860948, 9.094868849623160453923888358846, 9.539813252355737374323590509096, 10.27521243922981499066926751279

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