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Properties

Label 2-935-935.934-c0-0-6
Degree $2$
Conductor $935$
Sign $1$
Analytic cond. $0.466625$
Root an. cond. $0.683100$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.24·2-s − 1.80·3-s + 0.554·4-s + 5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s + 1.24·10-s + 11-s − 0.999·12-s − 0.445·13-s − 1.80·15-s − 1.24·16-s + 17-s + 2.80·18-s + 0.554·20-s + 1.24·22-s + 1.24·23-s + 1.00·24-s + 25-s − 0.554·26-s − 2.24·27-s + 1.24·29-s − 2.24·30-s − 0.999·32-s − 1.80·33-s + 1.24·34-s + ⋯
L(s)  = 1  + 1.24·2-s − 1.80·3-s + 0.554·4-s + 5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s + 1.24·10-s + 11-s − 0.999·12-s − 0.445·13-s − 1.80·15-s − 1.24·16-s + 17-s + 2.80·18-s + 0.554·20-s + 1.24·22-s + 1.24·23-s + 1.00·24-s + 25-s − 0.554·26-s − 2.24·27-s + 1.24·29-s − 2.24·30-s − 0.999·32-s − 1.80·33-s + 1.24·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.466625\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (934, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249072742\)
\(L(\frac12)\) \(\approx\) \(1.249072742\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 - 1.24T + T^{2} \)
3 \( 1 + 1.80T + T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + 0.445T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.24T + T^{2} \)
29 \( 1 - 1.24T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.445T + T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
43 \( 1 + 1.80T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.445T + T^{2} \)
61 \( 1 + 0.445T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + 0.445T + T^{2} \)
89 \( 1 + 1.80T + T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−10.37034089435210924279773653500, −9.814606012275510385987571716707, −8.775153779904487781157897685312, −6.92241432493231849511107776301, −6.59402197492388181289548226956, −5.67620686683104308804956930482, −5.15829608918138763544431478266, −4.47020435395080828835899950712, −3.16809467077707769060617953810, −1.39766954077805214544554947556, 1.39766954077805214544554947556, 3.16809467077707769060617953810, 4.47020435395080828835899950712, 5.15829608918138763544431478266, 5.67620686683104308804956930482, 6.59402197492388181289548226956, 6.92241432493231849511107776301, 8.775153779904487781157897685312, 9.814606012275510385987571716707, 10.37034089435210924279773653500

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