Properties

Label 2-950-1.1-c1-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.41·3-s + 4-s − 2.41·6-s − 1.58·7-s + 8-s + 2.82·9-s + 1.41·11-s − 2.41·12-s − 0.171·13-s − 1.58·14-s + 16-s − 17-s + 2.82·18-s − 19-s + 3.82·21-s + 1.41·22-s − 9.24·23-s − 2.41·24-s − 0.171·26-s + 0.414·27-s − 1.58·28-s − 5.82·29-s − 2.24·31-s + 32-s − 3.41·33-s − 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.985·6-s − 0.599·7-s + 0.353·8-s + 0.942·9-s + 0.426·11-s − 0.696·12-s − 0.0475·13-s − 0.423·14-s + 0.250·16-s − 0.242·17-s + 0.666·18-s − 0.229·19-s + 0.835·21-s + 0.301·22-s − 1.92·23-s − 0.492·24-s − 0.0336·26-s + 0.0797·27-s − 0.299·28-s − 1.08·29-s − 0.402·31-s + 0.176·32-s − 0.594·33-s − 0.171·34-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: \(1\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 3T^{2} \)
7 \( 1 + 1.58T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + 0.171T + 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
23 \( 1 + 9.24T + 23T^{2} \)
29 \( 1 + 5.82T + 29T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 5.75T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 5.48T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−9.990882895247461372351482260707, −8.873214855841981064270481712128, −7.60788726526879247062694078980, −6.66322976429905311776263874378, −6.08661881876646631595050138314, −5.38621816385852542083535268864, −4.38463132430675439502205353518, −3.46644205862699798920171307809, −1.85947418805391676913251396052, 0, 1.85947418805391676913251396052, 3.46644205862699798920171307809, 4.38463132430675439502205353518, 5.38621816385852542083535268864, 6.08661881876646631595050138314, 6.66322976429905311776263874378, 7.60788726526879247062694078980, 8.873214855841981064270481712128, 9.990882895247461372351482260707

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