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Properties

Label 2-122-61.20-c1-0-4
Degree $2$
Conductor $122$
Sign $0.478 + 0.878i$
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 + (0.0749 + 0.230i)3-s + (−0.809 + 0.587i)4-s + (1.59 − 1.15i)5-s + (0.196 − 0.142i)6-s + (0.425 − 1.30i)7-s + (0.809 + 0.587i)8-s + (2.37 − 1.72i)9-s + (−1.59 − 1.15i)10-s − 3.19·11-s + (−0.196 − 0.142i)12-s + 3.05·13-s − 1.37·14-s + (0.386 + 0.280i)15-s + (0.309 − 0.951i)16-s + (−5.15 + 3.74i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.0432 + 0.133i)3-s + (−0.404 + 0.293i)4-s + (0.711 − 0.517i)5-s + (0.0801 − 0.0582i)6-s + (0.160 − 0.494i)7-s + (0.286 + 0.207i)8-s + (0.793 − 0.576i)9-s + (−0.503 − 0.365i)10-s − 0.962·11-s + (−0.0566 − 0.0411i)12-s + 0.846·13-s − 0.367·14-s + (0.0997 + 0.0724i)15-s + (0.0772 − 0.237i)16-s + (−1.25 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(122\)    =    \(2 \cdot 61\)
Sign: $0.478 + 0.878i$
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: \((2,\ 122,\ (\ :1/2),\ 0.478 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897400 - 0.533079i\)
\(L(\frac12)\) \(\approx\) \(0.897400 - 0.533079i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.71 + 6.86i)T \)
good3 \( 1 + (-0.0749 - 0.230i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.59 + 1.15i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.425 + 1.30i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + 3.19T + 11T^{2} \)
13 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 + (5.15 - 3.74i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.628 - 1.93i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.156 - 0.113i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 - 2.56T + 29T^{2} \)
31 \( 1 + (2.70 - 8.31i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.71 - 5.27i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.65 + 8.17i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.33 + 2.42i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + 8.97T + 47T^{2} \)
53 \( 1 + (3.95 + 2.87i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.931 + 2.86i)T + (-47.7 + 34.6i)T^{2} \)
67 \( 1 + (-7.53 + 5.47i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.87 + 2.09i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.52 + 4.74i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.9 - 7.95i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.78 - 5.49i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-4.16 - 12.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.41 + 4.35i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.13584104224119142493142232684, −12.43678551162429948012786117019, −10.92392826764159595493983954724, −10.26114815326428059937623143130, −9.186326419727115458961868802204, −8.194902575709217943647301764955, −6.62647640657842901043133311518, −5.05184590244270743017704706808, −3.70056932561394178400053416295, −1.63176921683906889868275936446, 2.35517740926777704355766666241, 4.69000964488635288149761168859, 5.94430292287539335018120088927, 7.03526728517985270602435133040, 8.152994001248841718067642708265, 9.350261748086481373720740531160, 10.36890284069722082195407436318, 11.35462686491721229106322161334, 13.16945403013522818679237116511, 13.45525647038885183329793227179

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