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Properties

Label 2-62-31.8-c1-0-1
Degree $2$
Conductor $62$
Sign $0.989 + 0.141i$
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 + (0.649 + 1.99i)3-s + (−0.809 − 0.587i)4-s + 0.681·5-s + 2.10·6-s + (−2.28 − 1.65i)7-s + (−0.809 + 0.587i)8-s + (−1.14 + 0.835i)9-s + (0.210 − 0.648i)10-s + (−3.87 − 2.81i)11-s + (0.649 − 1.99i)12-s + (0.978 + 3.01i)13-s + (−2.28 + 1.65i)14-s + (0.442 + 1.36i)15-s + (0.309 + 0.951i)16-s + (2.33 − 1.69i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.375 + 1.15i)3-s + (−0.404 − 0.293i)4-s + 0.304·5-s + 0.858·6-s + (−0.863 − 0.627i)7-s + (−0.286 + 0.207i)8-s + (−0.383 + 0.278i)9-s + (0.0665 − 0.204i)10-s + (−1.16 − 0.847i)11-s + (0.187 − 0.577i)12-s + (0.271 + 0.835i)13-s + (−0.610 + 0.443i)14-s + (0.114 + 0.351i)15-s + (0.0772 + 0.237i)16-s + (0.566 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62\)    =    \(2 \cdot 31\)
Sign: $0.989 + 0.141i$
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: \((2,\ 62,\ (\ :1/2),\ 0.989 + 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994849 - 0.0707918i\)
\(L(\frac12)\) \(\approx\) \(0.994849 - 0.0707918i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.02 + 3.84i)T \)
good3 \( 1 + (-0.649 - 1.99i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 - 0.681T + 5T^{2} \)
7 \( 1 + (2.28 + 1.65i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (3.87 + 2.81i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.978 - 3.01i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.33 + 1.69i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.34 - 7.20i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.31 + 2.41i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.71 + 5.26i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 5.44T + 37T^{2} \)
41 \( 1 + (2.81 - 8.65i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.98 - 9.18i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.07 + 3.32i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.73 + 1.98i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.870 + 2.67i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 - 2.70T + 67T^{2} \)
71 \( 1 + (5.58 - 4.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.32 + 0.966i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (6.70 - 4.86i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.92 + 15.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (8.11 + 5.89i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.366 + 0.266i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.89517264561846974265958468293, −13.80723903623013823459177847052, −12.97128350028939206369798781224, −11.39461059858704449432495617986, −10.10286970450313245569671320324, −9.806826129549161043286819878709, −8.249575546877155783841995545921, −6.06342779638369951582640801206, −4.35913510148740032391222036256, −3.14034038110398688374794328790, 2.72515917977716506852677048302, 5.31825142155208792642946489868, 6.67326917599417497162834425540, 7.66565742878375175363866173193, 8.875352300182496075913374365787, 10.31657856393532733925456625994, 12.36792642431234066171441579972, 12.99427014000384783037744780931, 13.65954656013052659096074398598, 15.21464894537190691280065809352

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