Properties

Label 2-350-1.1-c1-0-7
Degree $2$
Conductor $350$
Sign $-1$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 7-s − 8-s − 2·9-s + 3·11-s − 12-s − 2·13-s + 14-s + 16-s − 3·17-s + 2·18-s − 7·19-s + 21-s − 3·22-s + 24-s + 2·26-s + 5·27-s − 28-s − 6·29-s − 4·31-s − 32-s − 3·33-s + 3·34-s − 2·36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.60·19-s + 0.218·21-s − 0.639·22-s + 0.204·24-s + 0.392·26-s + 0.962·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.97618644248142866152725079598, −10.18616620126561124117671237886, −9.056880838216350511563756479245, −8.489865387400252022087843283105, −7.02864082573576697328686349834, −6.40152272675796392235796567725, −5.26770803610804114861303773339, −3.76060769659553506276694871497, −2.12269688406492296074462119899, 0, 2.12269688406492296074462119899, 3.76060769659553506276694871497, 5.26770803610804114861303773339, 6.40152272675796392235796567725, 7.02864082573576697328686349834, 8.489865387400252022087843283105, 9.056880838216350511563756479245, 10.18616620126561124117671237886, 10.97618644248142866152725079598

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