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Properties

Label 2-62-31.2-c1-0-0
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: $\chi_{62} (33, \cdot )$
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

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

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