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Properties

Label 2-61-61.60-c7-0-16
Degree $2$
Conductor $61$
Sign $0.938 - 0.345i$
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  + 21.1i·2-s + 5.12·3-s − 320.·4-s − 325.·5-s + 108. i·6-s − 213. i·7-s − 4.07e3i·8-s − 2.16e3·9-s − 6.89e3i·10-s + 7.38e3i·11-s − 1.64e3·12-s + 9.94e3·13-s + 4.52e3·14-s − 1.66e3·15-s + 4.52e4·16-s − 1.18e4i·17-s + ⋯
L(s)  = 1  + 1.87i·2-s + 0.109·3-s − 2.50·4-s − 1.16·5-s + 0.205i·6-s − 0.235i·7-s − 2.81i·8-s − 0.987·9-s − 2.17i·10-s + 1.67i·11-s − 0.274·12-s + 1.25·13-s + 0.440·14-s − 0.127·15-s + 2.76·16-s − 0.585i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.938 - 0.345i$
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: \((2,\ 61,\ (\ :7/2),\ 0.938 - 0.345i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.468807 + 0.0836333i\)
\(L(\frac12)\) \(\approx\) \(0.468807 + 0.0836333i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-1.66e6 + 6.13e5i)T \)
good2 \( 1 - 21.1iT - 128T^{2} \)
3 \( 1 - 5.12T + 2.18e3T^{2} \)
5 \( 1 + 325.T + 7.81e4T^{2} \)
7 \( 1 + 213. iT - 8.23e5T^{2} \)
11 \( 1 - 7.38e3iT - 1.94e7T^{2} \)
13 \( 1 - 9.94e3T + 6.27e7T^{2} \)
17 \( 1 + 1.18e4iT - 4.10e8T^{2} \)
19 \( 1 - 2.01e4T + 8.93e8T^{2} \)
23 \( 1 + 4.06e4iT - 3.40e9T^{2} \)
29 \( 1 + 5.88e4iT - 1.72e10T^{2} \)
31 \( 1 + 6.65e4iT - 2.75e10T^{2} \)
37 \( 1 + 4.24e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.24e5T + 1.94e11T^{2} \)
43 \( 1 - 9.62e5iT - 2.71e11T^{2} \)
47 \( 1 + 9.56e5T + 5.06e11T^{2} \)
53 \( 1 + 1.64e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.49e6iT - 2.48e12T^{2} \)
67 \( 1 + 1.51e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.17e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.16e6T + 1.10e13T^{2} \)
79 \( 1 + 5.94e5iT - 1.92e13T^{2} \)
83 \( 1 + 8.84e6T + 2.71e13T^{2} \)
89 \( 1 - 6.40e6iT - 4.42e13T^{2} \)
97 \( 1 - 7.24e6T + 8.07e13T^{2} \)
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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

−14.12776020578540393675931174696, −12.91395486474399895559015827253, −11.50421090025122420723294137377, −9.607339275761300809396210166020, −8.375151643005705541431294049193, −7.60861687760264608997640854552, −6.52779763428086526807777670458, −5.01972477223717111756101924971, −3.83149389119080453136153771357, −0.21863164287177594356059657913, 1.08891031568429997530919115016, 3.13123044130104466019269522847, 3.71979474151109233507721947224, 5.57835925876458759099210362461, 8.361745871581006058139719533052, 8.764602735713734841279089371192, 10.54402622103863814691923839795, 11.47236275825930619125916322890, 11.82013287482649924607112788547, 13.33984802417825415390324556479

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