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Properties

Label 2-62-31.5-c1-0-1
Degree $2$
Conductor $62$
Sign $0.695 - 0.718i$
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  + 2-s + (−1.5 + 2.59i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−1.5 + 2.59i)6-s + (1.5 − 2.59i)7-s + 8-s + (−3 − 5.19i)9-s + (−0.5 − 0.866i)10-s + (1.5 + 2.59i)11-s + (−1.5 + 2.59i)12-s + (−2.5 − 4.33i)13-s + (1.5 − 2.59i)14-s + 3·15-s + 16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 + 1.49i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.612 + 1.06i)6-s + (0.566 − 0.981i)7-s + 0.353·8-s + (−1 − 1.73i)9-s + (−0.158 − 0.273i)10-s + (0.452 + 0.783i)11-s + (−0.433 + 0.749i)12-s + (−0.693 − 1.20i)13-s + (0.400 − 0.694i)14-s + 0.774·15-s + 0.250·16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62\)    =    \(2 \cdot 31\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(0.495072\)
Root analytic conductor: \(0.703613\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{62} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 62,\ (\ :1/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925492 + 0.392519i\)
\(L(\frac12)\) \(\approx\) \(0.925492 + 0.392519i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
31 \( 1 + (-2 + 5.19i)T \)
good3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.5 + 4.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−15.07215087684373906727504353545, −14.49586798721203059868545651464, −12.73513252130305159402070699571, −11.76314149804396347852035739965, −10.54210230380839543941245377517, −9.998128682271651124720912425896, −8.001071386375624658516722625264, −6.10879142913240382362487661533, −4.69882611932801093416359433427, −4.03457168001769038398374929217, 2.26389338676405307833668279124, 5.01827056275199137447909471844, 6.36717267283488216493841864950, 7.15663822429982815627228701898, 8.718211811453499604428634161231, 11.18846220196795281328918238964, 11.64340725718200534139312562372, 12.51084467933259813764398310948, 13.69140718218774055013063980316, 14.54671575955381139397634967053

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