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Properties

Label 2-124-124.123-c1-0-9
Degree $2$
Conductor $124$
Sign $0.118 + 0.992i$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0837 − 1.41i)2-s + (−1.98 + 0.236i)4-s + 3.80·5-s − 3.29i·7-s + (0.499 + 2.78i)8-s − 3·9-s + (−0.318 − 5.37i)10-s + (−4.65 + 0.275i)14-s + (3.88 − 0.938i)16-s + (0.251 + 4.23i)18-s + 7.99i·19-s + (−7.55 + 0.899i)20-s + 9.47·25-s + (0.779 + 6.54i)28-s + 5.56i·31-s + (−1.65 − 5.41i)32-s + ⋯
L(s)  = 1  + (−0.0592 − 0.998i)2-s + (−0.992 + 0.118i)4-s + 1.70·5-s − 1.24i·7-s + (0.176 + 0.984i)8-s − 9-s + (−0.100 − 1.69i)10-s + (−1.24 + 0.0737i)14-s + (0.972 − 0.234i)16-s + (0.0592 + 0.998i)18-s + 1.83i·19-s + (−1.68 + 0.201i)20-s + 1.89·25-s + (0.147 + 1.23i)28-s + 0.999i·31-s + (−0.291 − 0.956i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.118 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.839335 - 0.745353i\)
\(L(\frac12)\) \(\approx\) \(0.839335 - 0.745353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0837 + 1.41i)T \)
31 \( 1 - 5.56iT \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 3.80T + 5T^{2} \)
7 \( 1 + 3.29iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 7.99iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.5iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 19.6T + 97T^{2} \)
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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

−13.25971956259724924124028994299, −12.19414910367656663183890796559, −10.80230620472577489583912562076, −10.21433365489493738579696337550, −9.330032911167282455646489695768, −8.093822720284546744948258633827, −6.26186825514484817235647295836, −5.09014230766588803961707150480, −3.35666869492840259854001146037, −1.67992813549771985314553470647, 2.55155734352382594554209006106, 5.17237416699525419476560873108, 5.78312674115122735916530698104, 6.78325113390828498702889933087, 8.641698385033640643262341438669, 9.094072848607769739660614466637, 10.14101768811349374015357671513, 11.68973103470477511767434864691, 13.13414236681585805208165725852, 13.66996017350924322094269098402

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