Properties

Label 2-116-116.111-c0-0-0
Degree $2$
Conductor $116$
Sign $0.724 - 0.689i$
Analytic cond. $0.0578915$
Root an. cond. $0.240606$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−1.12 − 1.40i)5-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.400 − 1.75i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s + 1.24·17-s + (−0.900 − 0.433i)18-s + (1.62 − 0.781i)20-s + (−0.500 + 2.19i)25-s + (0.0990 + 0.433i)26-s + (−0.900 − 0.433i)29-s + (−0.222 − 0.974i)32-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−1.12 − 1.40i)5-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.400 − 1.75i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s + 1.24·17-s + (−0.900 − 0.433i)18-s + (1.62 − 0.781i)20-s + (−0.500 + 2.19i)25-s + (0.0990 + 0.433i)26-s + (−0.900 − 0.433i)29-s + (−0.222 − 0.974i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.724 - 0.689i$
Analytic conductor: \(0.0578915\)
Root analytic conductor: \(0.240606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :0),\ 0.724 - 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6567988611\)
\(L(\frac12)\) \(\approx\) \(0.6567988611\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 - 1.24T + T^{2} \)
19 \( 1 + (0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
41 \( 1 + 0.445T + T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 + (-0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
79 \( 1 + (-0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
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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

−13.93728280698268691617559719340, −12.92393525514533243599791940597, −12.08547159155992900167548059065, −11.31463289792714068287782596293, −9.209130155494114159252985183962, −8.262113482893172886022719944456, −7.61735796559339943625723157582, −5.80485425628311094218152912818, −4.81320037843895081907449787425, −3.57233705586428297510806529680, 2.97731371320272861811544350296, 3.78211432159173617962086038358, 5.66283865437418854723733575503, 6.89243826915145120674413000352, 8.279589527838960030130262015903, 9.852576512744813937403095779817, 10.93232029820198115034421540498, 11.53572814064135719676854503464, 12.37507573823034148052968421453, 13.79756665727488847040209539682

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