↑

Properties

Label 2-61-61.60-c7-0-6
Degree $2$
Conductor $61$
Sign $0.998 - 0.0626i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 19.5i·2-s + 32.3·3-s − 255.·4-s + 177.·5-s − 633. i·6-s + 1.32e3i·7-s + 2.50e3i·8-s − 1.14e3·9-s − 3.47e3i·10-s + 7.91e3i·11-s − 8.26e3·12-s − 1.03e3·13-s + 2.60e4·14-s + 5.73e3·15-s + 1.62e4·16-s + 3.36e4i·17-s + ⋯
L(s)  = 1  − 1.73i·2-s + 0.691·3-s − 1.99·4-s + 0.635·5-s − 1.19i·6-s + 1.46i·7-s + 1.72i·8-s − 0.522·9-s − 1.09i·10-s + 1.79i·11-s − 1.38·12-s − 0.130·13-s + 2.53·14-s + 0.438·15-s + 0.991·16-s + 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0626i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.998 - 0.0626i$
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: \((2,\ 61,\ (\ :7/2),\ 0.998 - 0.0626i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.54985 + 0.0486032i\)
\(L(\frac12)\) \(\approx\) \(1.54985 + 0.0486032i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-1.76e6 + 1.11e5i)T \)
good2 \( 1 + 19.5iT - 128T^{2} \)
3 \( 1 - 32.3T + 2.18e3T^{2} \)
5 \( 1 - 177.T + 7.81e4T^{2} \)
7 \( 1 - 1.32e3iT - 8.23e5T^{2} \)
11 \( 1 - 7.91e3iT - 1.94e7T^{2} \)
13 \( 1 + 1.03e3T + 6.27e7T^{2} \)
17 \( 1 - 3.36e4iT - 4.10e8T^{2} \)
19 \( 1 + 8.78e3T + 8.93e8T^{2} \)
23 \( 1 + 1.05e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.70e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.21e5iT - 2.75e10T^{2} \)
37 \( 1 + 3.71e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.27e5T + 1.94e11T^{2} \)
43 \( 1 - 3.53e5iT - 2.71e11T^{2} \)
47 \( 1 + 6.31e5T + 5.06e11T^{2} \)
53 \( 1 + 1.17e6iT - 1.17e12T^{2} \)
59 \( 1 - 2.64e6iT - 2.48e12T^{2} \)
67 \( 1 - 4.09e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.82e4iT - 9.09e12T^{2} \)
73 \( 1 - 4.59e6T + 1.10e13T^{2} \)
79 \( 1 - 4.35e6iT - 1.92e13T^{2} \)
83 \( 1 + 1.83e5T + 2.71e13T^{2} \)
89 \( 1 - 2.26e6iT - 4.42e13T^{2} \)
97 \( 1 + 3.73e6T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.08909016201451023767448600612, −12.46923419505958603280905129098, −11.44493138341703459824881084111, −9.979164389712346070322918834019, −9.348864510198315633895197728151, −8.269721088012032438329195057776, −5.77638207811900322409869436514, −4.11222952576755188737521687059, −2.35154015736402937774942090404, −2.05160026437231199019842218888, 0.50306164026083925252004678050, 3.36424876837539646994040381026, 5.10824105535804930536575006878, 6.35839475919828107898869704670, 7.50788688719761306290007817651, 8.518507730685952226504661400442, 9.514801949401178501834030180700, 11.07299191937808077860382224904, 13.45472957752613827177032650424, 13.91416180684905955355237732342

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