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Properties

Label 2-62-31.16-c1-0-3
Degree $2$
Conductor $62$
Sign $-0.632 + 0.774i$
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  + (−0.809 + 0.587i)2-s + (−2.48 − 1.80i)3-s + (0.309 − 0.951i)4-s − 3.34·5-s + 3.06·6-s + (0.778 − 2.39i)7-s + (0.309 + 0.951i)8-s + (1.98 + 6.10i)9-s + (2.70 − 1.96i)10-s + (0.532 − 1.63i)11-s + (−2.48 + 1.80i)12-s + (−1.89 − 1.37i)13-s + (0.778 + 2.39i)14-s + (8.30 + 6.03i)15-s + (−0.809 − 0.587i)16-s + (−0.244 − 0.751i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−1.43 − 1.04i)3-s + (0.154 − 0.475i)4-s − 1.49·5-s + 1.25·6-s + (0.294 − 0.905i)7-s + (0.109 + 0.336i)8-s + (0.660 + 2.03i)9-s + (0.856 − 0.622i)10-s + (0.160 − 0.493i)11-s + (−0.716 + 0.520i)12-s + (−0.524 − 0.381i)13-s + (0.208 + 0.640i)14-s + (2.14 + 1.55i)15-s + (−0.202 − 0.146i)16-s + (−0.0592 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62\)    =    \(2 \cdot 31\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(0.495072\)
Root analytic conductor: \(0.703613\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{62} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 62,\ (\ :1/2),\ -0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.110453 - 0.232790i\)
\(L(\frac12)\) \(\approx\) \(0.110453 - 0.232790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (2.94 + 4.72i)T \)
good3 \( 1 + (2.48 + 1.80i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 3.34T + 5T^{2} \)
7 \( 1 + (-0.778 + 2.39i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.532 + 1.63i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.89 + 1.37i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.244 + 0.751i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.326 - 0.237i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.56 + 4.82i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.20 + 3.05i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 5.88T + 37T^{2} \)
41 \( 1 + (6.27 - 4.55i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (6.18 - 4.49i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-3.91 - 2.84i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.812 + 2.50i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.53 - 2.56i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 9.60T + 61T^{2} \)
67 \( 1 + 8.86T + 67T^{2} \)
71 \( 1 + (3.67 + 11.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.42 + 7.46i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.647 + 1.99i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-9.91 + 7.20i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.52 + 13.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.80 - 17.8i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.84105808829343827128651832584, −13.32118598358311364819476276825, −12.04830870113434911594594002346, −11.35322138343677216334297768293, −10.42839831957398250364652229503, −8.084713337209179841156464133380, −7.42762993573266940455642050829, −6.32879942218616763432580163970, −4.62471239985438708758071791465, −0.52794830710636374841122311341, 3.90876293882549813645689938395, 5.17421994580105140924874448250, 7.02433093754683480321980441038, 8.641589770895490413279627901432, 9.925928103569091533928827566888, 11.07852658016628757061879254060, 11.87654670486565723422814686458, 12.27454015721469397683425761836, 14.97599187101665572405420121795, 15.64379536778173587906620411901

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