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Properties

Label 2-61-61.48-c5-0-15
Degree $2$
Conductor $61$
Sign $0.660 - 0.750i$
Analytic cond. $9.78341$
Root an. cond. $3.12784$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.75 + 3.90i)2-s + 18.1·3-s + (14.4 + 25.0i)4-s + (3.74 − 6.48i)5-s + (122. + 70.7i)6-s + (84.2 + 48.6i)7-s − 24.1i·8-s + 85.6·9-s + (50.6 − 29.2i)10-s + 687. i·11-s + (261. + 453. i)12-s + (497. − 861. i)13-s + (379. + 657. i)14-s + (67.9 − 117. i)15-s + (556. − 964. i)16-s + (−1.89e3 + 1.09e3i)17-s + ⋯
L(s)  = 1  + (1.19 + 0.689i)2-s + 1.16·3-s + (0.451 + 0.782i)4-s + (0.0670 − 0.116i)5-s + (1.38 + 0.802i)6-s + (0.649 + 0.374i)7-s − 0.133i·8-s + 0.352·9-s + (0.160 − 0.0924i)10-s + 1.71i·11-s + (0.525 + 0.909i)12-s + (0.816 − 1.41i)13-s + (0.517 + 0.896i)14-s + (0.0779 − 0.134i)15-s + (0.543 − 0.941i)16-s + (−1.59 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.660 - 0.750i$
Analytic conductor: \(9.78341\)
Root analytic conductor: \(3.12784\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :5/2),\ 0.660 - 0.750i)\)

Particular Values

\(L(3)\) \(\approx\) \(4.06512 + 1.83666i\)
\(L(\frac12)\) \(\approx\) \(4.06512 + 1.83666i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (2.82e4 + 6.94e3i)T \)
good2 \( 1 + (-6.75 - 3.90i)T + (16 + 27.7i)T^{2} \)
3 \( 1 - 18.1T + 243T^{2} \)
5 \( 1 + (-3.74 + 6.48i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-84.2 - 48.6i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 687. iT - 1.61e5T^{2} \)
13 \( 1 + (-497. + 861. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (1.89e3 - 1.09e3i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (892. + 1.54e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 2.36e3iT - 6.43e6T^{2} \)
29 \( 1 + (3.87e3 - 2.23e3i)T + (1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-515. + 297. i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 9.24e3iT - 6.93e7T^{2} \)
41 \( 1 + 476.T + 1.15e8T^{2} \)
43 \( 1 + (-8.64e3 - 4.99e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (7.10e3 + 1.23e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 5.00e3iT - 4.18e8T^{2} \)
59 \( 1 + (2.14e3 + 1.24e3i)T + (3.57e8 + 6.19e8i)T^{2} \)
67 \( 1 + (-6.00e4 - 3.46e4i)T + (6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.78e4 + 2.18e4i)T + (9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-2.93e4 - 5.08e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.75e4 - 1.01e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (3.61e4 - 6.26e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 1.03e5iT - 5.58e9T^{2} \)
97 \( 1 + (-5.13e4 - 8.88e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−14.45140592686884499324241210972, −13.06639913346093982591772839585, −12.81435298431062253367144015862, −10.86027988899206234653894189577, −9.183695299543108619887065232213, −8.090526276417744600617679461660, −6.79055077750647906022217002285, −5.19525690452089592821357256352, −3.98445912771753929096232249456, −2.29999915281199801933511328248, 1.94259463076886951859238754584, 3.32431068368545937386700565137, 4.38093231333626694310569892334, 6.17797358404781963823295433821, 8.162598201746858881642477870443, 9.024849091487495378207453005854, 11.03837300911337290518215041665, 11.51846740781485783552405725949, 13.27489484717269370129690328294, 13.95783926261683955090597627719

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