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Properties

Label 2-61-61.60-c3-0-0
Degree $2$
Conductor $61$
Sign $0.360 - 0.932i$
Analytic cond. $3.59911$
Root an. cond. $1.89713$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.98i·2-s − 9.52·3-s − 0.919·4-s − 2.16·5-s + 28.4i·6-s + 27.0i·7-s − 21.1i·8-s + 63.7·9-s + 6.46i·10-s + 41.4i·11-s + 8.75·12-s − 74.8·13-s + 80.7·14-s + 20.6·15-s − 70.5·16-s + 66.4i·17-s + ⋯
L(s)  = 1  − 1.05i·2-s − 1.83·3-s − 0.114·4-s − 0.193·5-s + 1.93i·6-s + 1.46i·7-s − 0.934i·8-s + 2.36·9-s + 0.204i·10-s + 1.13i·11-s + 0.210·12-s − 1.59·13-s + 1.54·14-s + 0.355·15-s − 1.10·16-s + 0.947i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(3.59911\)
Root analytic conductor: \(1.89713\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :3/2),\ 0.360 - 0.932i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.320838 + 0.219924i\)
\(L(\frac12)\) \(\approx\) \(0.320838 + 0.219924i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-171. + 444. i)T \)
good2 \( 1 + 2.98iT - 8T^{2} \)
3 \( 1 + 9.52T + 27T^{2} \)
5 \( 1 + 2.16T + 125T^{2} \)
7 \( 1 - 27.0iT - 343T^{2} \)
11 \( 1 - 41.4iT - 1.33e3T^{2} \)
13 \( 1 + 74.8T + 2.19e3T^{2} \)
17 \( 1 - 66.4iT - 4.91e3T^{2} \)
19 \( 1 + 9.08T + 6.85e3T^{2} \)
23 \( 1 - 12.6iT - 1.21e4T^{2} \)
29 \( 1 + 162. iT - 2.43e4T^{2} \)
31 \( 1 - 278. iT - 2.97e4T^{2} \)
37 \( 1 + 67.6iT - 5.06e4T^{2} \)
41 \( 1 - 129.T + 6.89e4T^{2} \)
43 \( 1 + 156. iT - 7.95e4T^{2} \)
47 \( 1 - 28.7T + 1.03e5T^{2} \)
53 \( 1 - 512. iT - 1.48e5T^{2} \)
59 \( 1 - 209. iT - 2.05e5T^{2} \)
67 \( 1 + 87.7iT - 3.00e5T^{2} \)
71 \( 1 + 453. iT - 3.57e5T^{2} \)
73 \( 1 + 1.02e3T + 3.89e5T^{2} \)
79 \( 1 - 893. iT - 4.93e5T^{2} \)
83 \( 1 + 282.T + 5.71e5T^{2} \)
89 \( 1 + 462. iT - 7.04e5T^{2} \)
97 \( 1 - 541.T + 9.12e5T^{2} \)
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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

−15.15098495212044661578776088480, −12.60142509019543103799224711064, −12.28533041185401983759700752687, −11.64849314138623414387011437940, −10.43766072239994878728704106800, −9.609373176033944907142393127052, −7.19547638472313796697485373067, −5.88003914298981094892108465179, −4.57439698832247732866492600393, −2.03574681642661671503271590183, 0.32208954726862760753064170696, 4.58660698648068957171921080685, 5.72760277061938893209568571818, 6.92325645268263692614469853998, 7.60787893581849806913406803089, 9.973830992802274432659229335458, 11.10608833568471675090356189484, 11.76885141590063781579491896555, 13.28939014320512430694222744317, 14.53347778784881787502723649791

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