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Properties

Label 2-950-19.16-c1-0-6
Degree $2$
Conductor $950$
Sign $0.580 - 0.813i$
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  + (−0.766 − 0.642i)2-s + (0.845 − 0.307i)3-s + (0.173 + 0.984i)4-s + (−0.845 − 0.307i)6-s + (1.06 + 1.84i)7-s + (0.500 − 0.866i)8-s + (−1.67 + 1.40i)9-s + (−0.575 + 0.996i)11-s + (0.449 + 0.779i)12-s + (2.12 + 0.773i)13-s + (0.370 − 2.10i)14-s + (−0.939 + 0.342i)16-s + (0.0280 + 0.0235i)17-s + 2.19·18-s + (−3.00 − 3.15i)19-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.488 − 0.177i)3-s + (0.0868 + 0.492i)4-s + (−0.345 − 0.125i)6-s + (0.403 + 0.698i)7-s + (0.176 − 0.306i)8-s + (−0.559 + 0.469i)9-s + (−0.173 + 0.300i)11-s + (0.129 + 0.224i)12-s + (0.589 + 0.214i)13-s + (0.0990 − 0.561i)14-s + (−0.234 + 0.0855i)16-s + (0.00680 + 0.00571i)17-s + 0.516·18-s + (−0.689 − 0.723i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.580 - 0.813i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.580 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06041 + 0.545976i\)
\(L(\frac12)\) \(\approx\) \(1.06041 + 0.545976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (3.00 + 3.15i)T \)
good3 \( 1 + (-0.845 + 0.307i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.06 - 1.84i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.575 - 0.996i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 - 0.773i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.0280 - 0.0235i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.40 - 7.97i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (4.74 - 3.98i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.21 - 3.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.64T + 37T^{2} \)
41 \( 1 + (7.81 - 2.84i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.121 + 0.687i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.57 + 4.68i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-2.05 - 11.6i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.27 - 1.06i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.03 + 5.86i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.48 - 6.28i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.70 - 9.66i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.53 + 2.01i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-8.50 + 3.09i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.73 + 4.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.65 - 2.42i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-6.34 - 5.32i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.14191771007709173522865230966, −9.011273322216544175589030707526, −8.776518222521507681685754119990, −7.83678577672513127907935716006, −7.08058243544570735064021721317, −5.81146412847503405385563838853, −4.89544208286535652038708656529, −3.54481047880567365449609021709, −2.54083154947573272134249412613, −1.60336944744952917262675358849, 0.63177234242974574556315939572, 2.27667444010634376640790128051, 3.60464822136752412874638322631, 4.53828711619422592738084674043, 5.86186121604158023426549298652, 6.47297300110844374720253978494, 7.65945887353918774143072733110, 8.283195079781233319069910308134, 8.883353246009259480247709607204, 9.833470192841589829893288255769

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