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Properties

Label 2-61-61.13-c1-0-2
Degree $2$
Conductor $61$
Sign $0.977 + 0.211i$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
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  + (0.198 − 0.343i)2-s − 0.716·3-s + (0.921 + 1.59i)4-s + (1.82 − 3.16i)5-s + (−0.141 + 0.245i)6-s + (−1.26 + 2.18i)7-s + 1.52·8-s − 2.48·9-s + (−0.723 − 1.25i)10-s − 4.36·11-s + (−0.659 − 1.14i)12-s + (0.443 − 0.768i)13-s + (0.5 + 0.866i)14-s + (−1.30 + 2.26i)15-s + (−1.54 + 2.66i)16-s + (2.14 + 3.71i)17-s + ⋯
L(s)  = 1  + (0.140 − 0.242i)2-s − 0.413·3-s + (0.460 + 0.798i)4-s + (0.816 − 1.41i)5-s + (−0.0579 + 0.100i)6-s + (−0.476 + 0.825i)7-s + 0.538·8-s − 0.829·9-s + (−0.228 − 0.396i)10-s − 1.31·11-s + (−0.190 − 0.329i)12-s + (0.123 − 0.213i)13-s + (0.133 + 0.231i)14-s + (−0.337 + 0.584i)15-s + (−0.385 + 0.667i)16-s + (0.520 + 0.900i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917015 - 0.0981035i\)
\(L(\frac12)\) \(\approx\) \(0.917015 - 0.0981035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (1.48 - 7.66i)T \)
good2 \( 1 + (-0.198 + 0.343i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 0.716T + 3T^{2} \)
5 \( 1 + (-1.82 + 3.16i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.26 - 2.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.36T + 11T^{2} \)
13 \( 1 + (-0.443 + 0.768i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.14 - 3.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.69 + 4.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.92T + 23T^{2} \)
29 \( 1 + (3.71 + 6.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.66 - 6.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.30T + 37T^{2} \)
41 \( 1 - 0.964T + 41T^{2} \)
43 \( 1 + (0.230 - 0.399i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.23 + 3.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.555T + 53T^{2} \)
59 \( 1 + (-1.91 + 3.32i)T + (-29.5 - 51.0i)T^{2} \)
67 \( 1 + (-1.15 + 2.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.47 + 2.55i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.96 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.20 - 2.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.751 + 1.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (-9.54 - 16.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.30264509179740810479608863256, −13.27777979781200800350737104246, −12.87606854080153356531454797656, −11.91270226342354892980908093400, −10.61927534705073003013700391609, −9.035019719379610533029883226835, −8.149174819376804837635903458281, −6.09404217365371217405058964973, −5.01701817674487427917293667213, −2.63298692183263453688449280118, 2.77701072045755931413291807549, 5.45415961530012677745525637633, 6.40528351660089812410186144925, 7.44400237127642694904206866982, 9.819681287894063279635517218856, 10.58636607404156602779137960531, 11.25801084752025723117556606903, 13.23073707535855399714452212104, 14.20489591600423237987943596648, 14.85450699413024212897111311646

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