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Properties

Label 2-61-61.60-c7-0-24
Degree $2$
Conductor $61$
Sign $0.968 - 0.247i$
Analytic cond. $19.0554$
Root an. cond. $4.36525$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.16i·2-s + 87.9·3-s + 89.9·4-s − 212.·5-s + 542. i·6-s − 1.69e3i·7-s + 1.34e3i·8-s + 5.55e3·9-s − 1.31e3i·10-s + 2.68e3i·11-s + 7.91e3·12-s + 7.59e3·13-s + 1.04e4·14-s − 1.86e4·15-s + 3.23e3·16-s − 1.86e4i·17-s + ⋯
L(s)  = 1  + 0.544i·2-s + 1.88·3-s + 0.703·4-s − 0.760·5-s + 1.02i·6-s − 1.86i·7-s + 0.928i·8-s + 2.53·9-s − 0.414i·10-s + 0.607i·11-s + 1.32·12-s + 0.958·13-s + 1.01·14-s − 1.43·15-s + 0.197·16-s − 0.919i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.968 - 0.247i$
Analytic conductor: \(19.0554\)
Root analytic conductor: \(4.36525\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :7/2),\ 0.968 - 0.247i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.96235 + 0.497494i\)
\(L(\frac12)\) \(\approx\) \(3.96235 + 0.497494i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-1.71e6 + 4.38e5i)T \)
good2 \( 1 - 6.16iT - 128T^{2} \)
3 \( 1 - 87.9T + 2.18e3T^{2} \)
5 \( 1 + 212.T + 7.81e4T^{2} \)
7 \( 1 + 1.69e3iT - 8.23e5T^{2} \)
11 \( 1 - 2.68e3iT - 1.94e7T^{2} \)
13 \( 1 - 7.59e3T + 6.27e7T^{2} \)
17 \( 1 + 1.86e4iT - 4.10e8T^{2} \)
19 \( 1 + 5.61e3T + 8.93e8T^{2} \)
23 \( 1 - 3.34e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.58e5iT - 1.72e10T^{2} \)
31 \( 1 - 5.40e4iT - 2.75e10T^{2} \)
37 \( 1 + 5.13e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.45e5T + 1.94e11T^{2} \)
43 \( 1 - 3.61e5iT - 2.71e11T^{2} \)
47 \( 1 + 5.69e5T + 5.06e11T^{2} \)
53 \( 1 - 9.76e5iT - 1.17e12T^{2} \)
59 \( 1 + 3.18e5iT - 2.48e12T^{2} \)
67 \( 1 - 4.29e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.01e6iT - 9.09e12T^{2} \)
73 \( 1 + 4.23e6T + 1.10e13T^{2} \)
79 \( 1 + 2.54e6iT - 1.92e13T^{2} \)
83 \( 1 + 8.64e6T + 2.71e13T^{2} \)
89 \( 1 + 1.01e7iT - 4.42e13T^{2} \)
97 \( 1 + 1.37e7T + 8.07e13T^{2} \)
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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

−13.98695862817520254286412545118, −12.94728146141429358657325027906, −11.16762609255616735777443409036, −9.996796661240687175253170498262, −8.513996574993687779639817889880, −7.41124278840002778525544332483, −7.14770321827676111720795859522, −4.23143794378584034548671221614, −3.23022423591968852781154584316, −1.47393400625485583881355948649, 1.79112581103343099589283060603, 2.82255975316286351109886245533, 3.79162774137334207112989543641, 6.32602118580851593596965628459, 8.110709608815041429044196543359, 8.579164322245257189434918650924, 9.873820836711071546253677680115, 11.41628886530027344245396037265, 12.42311723859464624621501559492, 13.43000449730660632138355598648

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