Properties

Degree $2$
Conductor $950$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s − 2·9-s + 2·11-s + 12-s + 13-s + 3·14-s + 16-s − 3·17-s + 2·18-s − 19-s − 3·21-s − 2·22-s + 23-s − 24-s − 26-s − 5·27-s − 3·28-s − 5·29-s − 8·31-s − 32-s + 2·33-s + 3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.654·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s − 0.566·28-s − 0.928·29-s − 1.43·31-s − 0.176·32-s + 0.348·33-s + 0.514·34-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: $-1$
Motivic weight: \(1\)
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 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.400115216603937442308106741391, −8.925019181414506634794762446017, −8.187366436155753750941904167188, −7.08084872197682116999638941334, −6.43803628048625394061789390103, −5.47809446086921756820160858625, −3.84969924867073870114275032751, −3.08844310830142045553246780849, −1.90909315273381969795467705091, 0, 1.90909315273381969795467705091, 3.08844310830142045553246780849, 3.84969924867073870114275032751, 5.47809446086921756820160858625, 6.43803628048625394061789390103, 7.08084872197682116999638941334, 8.187366436155753750941904167188, 8.925019181414506634794762446017, 9.400115216603937442308106741391

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