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Properties

Label 2-124-31.2-c1-0-1
Degree $2$
Conductor $124$
Sign $0.851 + 0.525i$
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  + (1.80 − 1.31i)3-s − 0.381·5-s + (0.0729 + 0.224i)7-s + (0.618 − 1.90i)9-s + (−0.381 − 1.17i)11-s + (−3.73 + 2.71i)13-s + (−0.690 + 0.502i)15-s + (0.690 − 2.12i)17-s + (5.42 + 3.94i)19-s + (0.427 + 0.310i)21-s + (−0.545 + 1.67i)23-s − 4.85·25-s + (0.690 + 2.12i)27-s + (−4.54 − 3.30i)29-s + (−4.19 + 3.66i)31-s + ⋯
L(s)  = 1  + (1.04 − 0.758i)3-s − 0.170·5-s + (0.0275 + 0.0848i)7-s + (0.206 − 0.634i)9-s + (−0.115 − 0.354i)11-s + (−1.03 + 0.752i)13-s + (−0.178 + 0.129i)15-s + (0.167 − 0.515i)17-s + (1.24 + 0.904i)19-s + (0.0931 + 0.0677i)21-s + (−0.113 + 0.349i)23-s − 0.970·25-s + (0.132 + 0.409i)27-s + (−0.844 − 0.613i)29-s + (−0.752 + 0.658i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.851 + 0.525i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.851 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29324 - 0.366877i\)
\(L(\frac12)\) \(\approx\) \(1.29324 - 0.366877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + (4.19 - 3.66i)T \)
good3 \( 1 + (-1.80 + 1.31i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
7 \( 1 + (-0.0729 - 0.224i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.381 + 1.17i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.73 - 2.71i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.690 + 2.12i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.42 - 3.94i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.545 - 1.67i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.54 + 3.30i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 + (8.47 + 6.15i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.927 + 0.673i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-6.92 + 5.03i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.83 - 8.73i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.2 + 7.46i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 3T + 67T^{2} \)
71 \( 1 + (-4.04 + 12.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.0450 - 0.138i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.35 + 4.61i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.95 + 15.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.45 - 4.47i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.60318016566813629747387917755, −12.34221357214069532668881736800, −11.55891560241883509095607332788, −9.931015782936032041508269101889, −8.983237240510302379593111486059, −7.79151135022666961191489077567, −7.16878497280932737513455594618, −5.43838059902636109936904563161, −3.56935255819313689782366702451, −2.07643553086583651340959732352, 2.70678949281703143112566840538, 3.99473777609185370911692397321, 5.35639575628121411714866196981, 7.27647888124592782538402290975, 8.214914697319898431383017073307, 9.458338073601617187507314253927, 10.02623521330768693392755135103, 11.36737021361213843571845508070, 12.60764058769592457123966377202, 13.65637726517266538816719608595

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