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Properties

Label 2-61-61.60-c7-0-1
Degree $2$
Conductor $61$
Sign $0.288 + 0.957i$
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  + 14.4i·2-s − 26.7·3-s − 81.2·4-s − 66.2·5-s − 386. i·6-s + 1.49e3i·7-s + 676. i·8-s − 1.47e3·9-s − 958. i·10-s − 1.16e3i·11-s + 2.16e3·12-s − 2.39e3·13-s − 2.15e4·14-s + 1.76e3·15-s − 2.01e4·16-s − 3.26e4i·17-s + ⋯
L(s)  = 1  + 1.27i·2-s − 0.571·3-s − 0.634·4-s − 0.237·5-s − 0.730i·6-s + 1.64i·7-s + 0.467i·8-s − 0.673·9-s − 0.303i·10-s − 0.263i·11-s + 0.362·12-s − 0.302·13-s − 2.10·14-s + 0.135·15-s − 1.23·16-s − 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.288 + 0.957i$
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: \((2,\ 61,\ (\ :7/2),\ 0.288 + 0.957i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.216275 - 0.160714i\)
\(L(\frac12)\) \(\approx\) \(0.216275 - 0.160714i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-5.11e5 - 1.69e6i)T \)
good2 \( 1 - 14.4iT - 128T^{2} \)
3 \( 1 + 26.7T + 2.18e3T^{2} \)
5 \( 1 + 66.2T + 7.81e4T^{2} \)
7 \( 1 - 1.49e3iT - 8.23e5T^{2} \)
11 \( 1 + 1.16e3iT - 1.94e7T^{2} \)
13 \( 1 + 2.39e3T + 6.27e7T^{2} \)
17 \( 1 + 3.26e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.35e4T + 8.93e8T^{2} \)
23 \( 1 - 4.67e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.25e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.42e5iT - 2.75e10T^{2} \)
37 \( 1 + 1.95e5iT - 9.49e10T^{2} \)
41 \( 1 - 1.48e5T + 1.94e11T^{2} \)
43 \( 1 + 9.21e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.04e6T + 5.06e11T^{2} \)
53 \( 1 + 8.30e5iT - 1.17e12T^{2} \)
59 \( 1 - 8.57e5iT - 2.48e12T^{2} \)
67 \( 1 - 2.14e6iT - 6.06e12T^{2} \)
71 \( 1 - 5.63e6iT - 9.09e12T^{2} \)
73 \( 1 + 5.99e6T + 1.10e13T^{2} \)
79 \( 1 + 1.83e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.12e6T + 2.71e13T^{2} \)
89 \( 1 - 6.17e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.67e7T + 8.07e13T^{2} \)
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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

−14.72918001221459093282961445952, −13.72680462617296998067917158223, −11.78274027026414197640328174839, −11.63759714320069848667988962159, −9.418705941654817087774229005045, −8.396635385610720004820912472278, −7.14189001020998114778418632056, −5.70497286689709862300012308822, −5.28644132169511347164204155618, −2.65695653210352552101805964386, 0.10857714216835469660571836041, 1.38438473543082948408709875277, 3.30849272324510376992847841666, 4.48985424532191520520020424425, 6.44936097066210816271217166836, 7.86149584557587283956250200378, 9.744975378232390351375805156912, 10.65795370725389646343983640300, 11.34803461330350997941105625704, 12.43127485526940687355475525858

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