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Properties

Label 2-62-31.8-c1-0-3
Degree $2$
Conductor $62$
Sign $0.118 + 0.992i$
Analytic cond. $0.495072$
Root an. cond. $0.703613$
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 + (−0.531 − 1.63i)3-s + (−0.809 − 0.587i)4-s − 1.68·5-s − 1.72·6-s + (3.90 + 2.83i)7-s + (−0.809 + 0.587i)8-s + (0.0317 − 0.0230i)9-s + (−0.519 + 1.59i)10-s + (1.13 + 0.824i)11-s + (−0.531 + 1.63i)12-s + (−0.933 − 2.87i)13-s + (3.90 − 2.83i)14-s + (0.894 + 2.75i)15-s + (0.309 + 0.951i)16-s + (−5.76 + 4.18i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.306 − 0.944i)3-s + (−0.404 − 0.293i)4-s − 0.751·5-s − 0.702·6-s + (1.47 + 1.07i)7-s + (−0.286 + 0.207i)8-s + (0.0105 − 0.00768i)9-s + (−0.164 + 0.505i)10-s + (0.342 + 0.248i)11-s + (−0.153 + 0.472i)12-s + (−0.258 − 0.796i)13-s + (1.04 − 0.757i)14-s + (0.230 + 0.710i)15-s + (0.0772 + 0.237i)16-s + (−1.39 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62\)    =    \(2 \cdot 31\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(0.495072\)
Root analytic conductor: \(0.703613\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{62} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 62,\ (\ :1/2),\ 0.118 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653376 - 0.579815i\)
\(L(\frac12)\) \(\approx\) \(0.653376 - 0.579815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-4.75 - 2.89i)T \)
good3 \( 1 + (0.531 + 1.63i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + 1.68T + 5T^{2} \)
7 \( 1 + (-3.90 - 2.83i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-1.13 - 0.824i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.933 + 2.87i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.76 - 4.18i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.15 - 3.56i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.225 + 0.164i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.980 + 3.01i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 3.08T + 37T^{2} \)
41 \( 1 + (-1.74 + 5.35i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-0.838 + 2.58i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (2.53 + 7.81i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.54 - 1.12i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.13 - 12.7i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 + 5.84T + 67T^{2} \)
71 \( 1 + (-0.153 + 0.111i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (12.0 + 8.76i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.6 + 7.75i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.46 + 10.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.0186 + 0.0135i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.45 - 2.51i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.84691343591709449998792834491, −13.35359559233243072290982927255, −12.14710518434155973111437647655, −11.85163449879380182808129054085, −10.60858672742274134585288541438, −8.694769349365105849254802511327, −7.74335929958145719841322336847, −5.98649536169807888639759264442, −4.38077387267054957572429698924, −1.93691816174177389332914676990, 4.33499688441599173565019927675, 4.68812954894910771822191929594, 6.92411157106339607658690577727, 8.018642432460351431146768707336, 9.414859428409754013464381282983, 11.04020535279676557387453846619, 11.49333207802047977814707549921, 13.45141810222322211965326748423, 14.39093179670177289049993056973, 15.38663393168025664220179671324

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