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Properties

Label 2-61-61.3-c1-0-0
Degree $2$
Conductor $61$
Sign $0.997 - 0.0655i$
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  + (−2.40 − 0.782i)2-s + (−0.277 + 0.853i)3-s + (3.56 + 2.59i)4-s + (0.429 + 0.311i)5-s + (1.33 − 1.83i)6-s + (3.57 − 1.16i)7-s + (−3.58 − 4.93i)8-s + (1.77 + 1.28i)9-s + (−0.789 − 1.08i)10-s + 3.16i·11-s + (−3.20 + 2.32i)12-s − 4.65·13-s − 9.52·14-s + (−0.385 + 0.279i)15-s + (2.04 + 6.29i)16-s + (3.59 − 4.94i)17-s + ⋯
L(s)  = 1  + (−1.70 − 0.553i)2-s + (−0.160 + 0.492i)3-s + (1.78 + 1.29i)4-s + (0.191 + 0.139i)5-s + (0.545 − 0.750i)6-s + (1.35 − 0.439i)7-s + (−1.26 − 1.74i)8-s + (0.591 + 0.429i)9-s + (−0.249 − 0.343i)10-s + 0.953i·11-s + (−0.924 + 0.671i)12-s − 1.29·13-s − 2.54·14-s + (−0.0994 + 0.0722i)15-s + (0.511 + 1.57i)16-s + (0.871 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.997 - 0.0655i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.997 - 0.0655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474296 + 0.0155672i\)
\(L(\frac12)\) \(\approx\) \(0.474296 + 0.0155672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (5.20 + 5.82i)T \)
good2 \( 1 + (2.40 + 0.782i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.277 - 0.853i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.429 - 0.311i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-3.57 + 1.16i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 - 3.16iT - 11T^{2} \)
13 \( 1 + 4.65T + 13T^{2} \)
17 \( 1 + (-3.59 + 4.94i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.859 - 2.64i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.52 - 2.09i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + 9.86iT - 29T^{2} \)
31 \( 1 + (0.188 - 0.0611i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.45 - 1.44i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.929 - 2.86i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.0652 - 0.0897i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + 7.12T + 47T^{2} \)
53 \( 1 + (-1.18 - 1.63i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.39 - 0.776i)T + (47.7 + 34.6i)T^{2} \)
67 \( 1 + (-5.54 + 7.63i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-2.69 - 3.70i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.80 - 2.04i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.93 + 5.41i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.572 + 1.76i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (11.9 + 3.88i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.87 - 5.76i)T + (-78.4 + 57.0i)T^{2} \)
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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

−15.42039121806856177588096290590, −14.16548536597264987736537495154, −12.19320829694116467051848585564, −11.34965654829665241415715428952, −10.00947298116178303780420815081, −9.845296263831873881036880566510, −7.951050635665186293289639792568, −7.36469296639521400740743503837, −4.71971889267101590411495049390, −2.01304563296478735404605422620, 1.59992046770254560072212279851, 5.51454372472980912282356997480, 6.98745189710268563353498543586, 8.027139139256957834556572339783, 8.955117638312667612998095884245, 10.24557122379688554477271845345, 11.33422688695326414786230358790, 12.52646342318035252013986624376, 14.46339813170602362487505624977, 15.22262361659219630997667833729

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