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Properties

Label 2-122-61.20-c1-0-2
Degree $2$
Conductor $122$
Sign $0.539 - 0.842i$
Analytic cond. $0.974174$
Root an. cond. $0.987002$
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  + (0.309 + 0.951i)2-s + (0.143 + 0.442i)3-s + (−0.809 + 0.587i)4-s + (2.83 − 2.06i)5-s + (−0.376 + 0.273i)6-s + (−1.39 + 4.28i)7-s + (−0.809 − 0.587i)8-s + (2.25 − 1.63i)9-s + (2.83 + 2.06i)10-s − 2.64·11-s + (−0.376 − 0.273i)12-s − 3.22·13-s − 4.50·14-s + (1.32 + 0.960i)15-s + (0.309 − 0.951i)16-s + (1.05 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.0830 + 0.255i)3-s + (−0.404 + 0.293i)4-s + (1.26 − 0.922i)5-s + (−0.153 + 0.111i)6-s + (−0.526 + 1.61i)7-s + (−0.286 − 0.207i)8-s + (0.750 − 0.545i)9-s + (0.897 + 0.652i)10-s − 0.797·11-s + (−0.108 − 0.0790i)12-s − 0.893·13-s − 1.20·14-s + (0.341 + 0.247i)15-s + (0.0772 − 0.237i)16-s + (0.255 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(122\)    =    \(2 \cdot 61\)
Sign: $0.539 - 0.842i$
Analytic conductor: \(0.974174\)
Root analytic conductor: \(0.987002\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{122} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 122,\ (\ :1/2),\ 0.539 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12539 + 0.615728i\)
\(L(\frac12)\) \(\approx\) \(1.12539 + 0.615728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-7.81 + 0.0477i)T \)
good3 \( 1 + (-0.143 - 0.442i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.83 + 2.06i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.39 - 4.28i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + 2.64T + 11T^{2} \)
13 \( 1 + 3.22T + 13T^{2} \)
17 \( 1 + (-1.05 + 0.766i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.40 + 4.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.40 + 2.47i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 + (-2.18 + 6.71i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.67 - 8.24i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.26 - 6.97i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (6.38 + 4.64i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + 0.945T + 47T^{2} \)
53 \( 1 + (-7.34 - 5.33i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.911 - 2.80i)T + (-47.7 + 34.6i)T^{2} \)
67 \( 1 + (9.84 - 7.15i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-7.41 - 5.38i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-7.77 - 5.65i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-6.09 - 4.42i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.862 + 2.65i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-1.23 - 3.80i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.70 + 8.32i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.31090098616936431534784962689, −12.92654660918484237947393433732, −11.97997855770107345148377562892, −9.916598797954939568907201602039, −9.373193265287504805922224758013, −8.504311838093724837229567007238, −6.76496923389595342498301492810, −5.59975038312940065174265569478, −4.87194811934401497443909563951, −2.56132111396414712897577940576, 1.95671704719888955687409126772, 3.52680588715595815303098549067, 5.21081859898121639115737756047, 6.74737007796332261592658925214, 7.58727742568072138733882562695, 9.662012896373328726565142039578, 10.36005075876949406283030308779, 10.72337145573414511657913432682, 12.59268086535701455293357761519, 13.35731097038735771694446615099

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