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Properties

Label 2-61-61.60-c7-0-25
Degree $2$
Conductor $61$
Sign $-0.351 + 0.936i$
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  − 0.698i·2-s + 31.4·3-s + 127.·4-s − 277.·5-s − 21.9i·6-s − 565. i·7-s − 178. i·8-s − 1.19e3·9-s + 193. i·10-s − 7.72e3i·11-s + 4.00e3·12-s − 8.47e3·13-s − 395.·14-s − 8.70e3·15-s + 1.61e4·16-s − 2.14e3i·17-s + ⋯
L(s)  = 1  − 0.0617i·2-s + 0.671·3-s + 0.996·4-s − 0.991·5-s − 0.0415i·6-s − 0.622i·7-s − 0.123i·8-s − 0.548·9-s + 0.0612i·10-s − 1.74i·11-s + 0.669·12-s − 1.06·13-s − 0.0384·14-s − 0.666·15-s + 0.988·16-s − 0.106i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $-0.351 + 0.936i$
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: \((2,\ 61,\ (\ :7/2),\ -0.351 + 0.936i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.998927 - 1.44182i\)
\(L(\frac12)\) \(\approx\) \(0.998927 - 1.44182i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (6.22e5 - 1.65e6i)T \)
good2 \( 1 + 0.698iT - 128T^{2} \)
3 \( 1 - 31.4T + 2.18e3T^{2} \)
5 \( 1 + 277.T + 7.81e4T^{2} \)
7 \( 1 + 565. iT - 8.23e5T^{2} \)
11 \( 1 + 7.72e3iT - 1.94e7T^{2} \)
13 \( 1 + 8.47e3T + 6.27e7T^{2} \)
17 \( 1 + 2.14e3iT - 4.10e8T^{2} \)
19 \( 1 - 3.23e4T + 8.93e8T^{2} \)
23 \( 1 - 3.04e4iT - 3.40e9T^{2} \)
29 \( 1 + 2.13e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.65e5iT - 2.75e10T^{2} \)
37 \( 1 + 4.95e3iT - 9.49e10T^{2} \)
41 \( 1 + 495.T + 1.94e11T^{2} \)
43 \( 1 - 8.25e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.99e5T + 5.06e11T^{2} \)
53 \( 1 + 7.45e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.80e6iT - 2.48e12T^{2} \)
67 \( 1 - 2.60e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.96e6iT - 9.09e12T^{2} \)
73 \( 1 - 6.43e5T + 1.10e13T^{2} \)
79 \( 1 + 8.12e5iT - 1.92e13T^{2} \)
83 \( 1 - 6.11e6T + 2.71e13T^{2} \)
89 \( 1 + 1.20e6iT - 4.42e13T^{2} \)
97 \( 1 + 3.70e6T + 8.07e13T^{2} \)
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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

−13.39466618329500359027418237530, −11.58283287005281660658892950591, −11.41241665110678652488647640780, −9.756601717485554829126375352774, −8.101929705426838466377944264415, −7.52064926216729660022213185522, −5.87133387600667529925587833830, −3.71889262559623173065204026888, −2.69981917579087803037989997712, −0.54880129154482764639570639475, 2.04774926614503655436035213953, 3.22831243026190452376372849342, 5.09563584215005040308874280938, 7.02745934680725283462379886214, 7.76135966738499860826253843293, 9.129628765441444565647223848480, 10.52893555805405378298078201222, 12.12165839179997879998617996990, 12.18838752476912745568491756417, 14.33797029278782807416651242646

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