Properties

Label 2-124-31.4-c1-0-0
Degree $2$
Conductor $124$
Sign $0.933 - 0.358i$
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  + (−0.190 + 0.587i)3-s + 2·5-s + (−0.309 + 0.224i)7-s + (2.11 + 1.53i)9-s + (−0.309 + 0.224i)11-s + (0.190 − 0.587i)13-s + (−0.381 + 1.17i)15-s + (−1.92 − 1.40i)17-s + (−0.572 − 1.76i)19-s + (−0.0729 − 0.224i)21-s + (−6.16 − 4.47i)23-s − 25-s + (−2.80 + 2.04i)27-s + (−1.80 − 5.56i)29-s + (−1.23 + 5.42i)31-s + ⋯
L(s)  = 1  + (−0.110 + 0.339i)3-s + 0.894·5-s + (−0.116 + 0.0848i)7-s + (0.706 + 0.512i)9-s + (−0.0931 + 0.0676i)11-s + (0.0529 − 0.163i)13-s + (−0.0986 + 0.303i)15-s + (−0.467 − 0.339i)17-s + (−0.131 − 0.404i)19-s + (−0.0159 − 0.0489i)21-s + (−1.28 − 0.933i)23-s − 0.200·25-s + (−0.540 + 0.392i)27-s + (−0.335 − 1.03i)29-s + (−0.222 + 0.975i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.933 - 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13776 + 0.210985i\)
\(L(\frac12)\) \(\approx\) \(1.13776 + 0.210985i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 5.42i)T \)
good3 \( 1 + (0.190 - 0.587i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (0.309 - 0.224i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.309 - 0.224i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.190 + 0.587i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.92 + 1.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.572 + 1.76i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (6.16 + 4.47i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.80 + 5.56i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 + (0.281 + 0.865i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.663 - 2.04i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-1.80 + 5.56i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.16 + 3.75i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.663 - 2.04i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (-8.78 - 6.37i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.39 - 6.10i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.30 - 1.67i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.04 + 9.37i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.92 - 4.30i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (11.6 - 8.45i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.45125980905639586435597857214, −12.63556529937289581089447068751, −11.26640760170377502825945775130, −10.19310414433890679352211815556, −9.546348042336218748692361613761, −8.172950037806157566288035276702, −6.77091952332651142697654887407, −5.56067872467934032443089727442, −4.28289565152854067390969419466, −2.23569078335373275558135236022, 1.85733808137285213790918834189, 3.93669002607716546537229718179, 5.68135150162046020823927506609, 6.61985374011884888005382683946, 7.87911980526500187428491479122, 9.362415443689986268240072028782, 10.05290589470534174010775603961, 11.33621126123688344150271577943, 12.52691846620564549844425130755, 13.28334789352040095609775042064

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