Properties

Label 2-950-1.1-c1-0-5
Degree $2$
Conductor $950$
Sign $1$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.03·3-s + 4-s − 3.03·6-s + 2.46·7-s + 8-s + 6.19·9-s + 0.728·11-s − 3.03·12-s − 6.23·13-s + 2.46·14-s + 16-s + 0.563·17-s + 6.19·18-s − 19-s − 7.49·21-s + 0.728·22-s + 4.63·23-s − 3.03·24-s − 6.23·26-s − 9.70·27-s + 2.46·28-s + 10.2·29-s + 6.06·31-s + 32-s − 2.21·33-s + 0.563·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.75·3-s + 0.5·4-s − 1.23·6-s + 0.933·7-s + 0.353·8-s + 2.06·9-s + 0.219·11-s − 0.875·12-s − 1.72·13-s + 0.660·14-s + 0.250·16-s + 0.136·17-s + 1.46·18-s − 0.229·19-s − 1.63·21-s + 0.155·22-s + 0.966·23-s − 0.619·24-s − 1.22·26-s − 1.86·27-s + 0.466·28-s + 1.89·29-s + 1.08·31-s + 0.176·32-s − 0.384·33-s + 0.0965·34-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.497478669\)
\(L(\frac12)\) \(\approx\) \(1.497478669\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 3.03T + 3T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 - 0.728T + 11T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 - 0.563T + 17T^{2} \)
23 \( 1 - 4.63T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 6.06T + 31T^{2} \)
37 \( 1 + 5.72T + 37T^{2} \)
41 \( 1 - 4.79T + 41T^{2} \)
43 \( 1 - 8.06T + 43T^{2} \)
47 \( 1 - 8.12T + 47T^{2} \)
53 \( 1 + 1.53T + 53T^{2} \)
59 \( 1 + 5.76T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 + 4.09T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 - 7.85T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−10.47812395760950593900892771064, −9.522129456197466936537817190141, −8.075894010530024812148858185685, −7.11204516473340676961782901409, −6.53049179238004494768702899980, −5.44986835051929002683014338050, −4.89242873982978485255593790771, −4.31480640408567750484737837572, −2.51140339553885340497874043869, −0.987570335650326101693547283034, 0.987570335650326101693547283034, 2.51140339553885340497874043869, 4.31480640408567750484737837572, 4.89242873982978485255593790771, 5.44986835051929002683014338050, 6.53049179238004494768702899980, 7.11204516473340676961782901409, 8.075894010530024812148858185685, 9.522129456197466936537817190141, 10.47812395760950593900892771064

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