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Properties

Label 2-61-61.60-c7-0-29
Degree $2$
Conductor $61$
Sign $-0.674 - 0.738i$
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  − 19.6i·2-s − 11.3·3-s − 259.·4-s + 547.·5-s + 223. i·6-s − 1.20e3i·7-s + 2.59e3i·8-s − 2.05e3·9-s − 1.07e4i·10-s − 1.97e3i·11-s + 2.94e3·12-s + 3.52e3·13-s − 2.36e4·14-s − 6.20e3·15-s + 1.78e4·16-s − 2.13e4i·17-s + ⋯
L(s)  = 1  − 1.74i·2-s − 0.242·3-s − 2.02·4-s + 1.95·5-s + 0.421i·6-s − 1.32i·7-s + 1.79i·8-s − 0.941·9-s − 3.40i·10-s − 0.446i·11-s + 0.491·12-s + 0.444·13-s − 2.30·14-s − 0.474·15-s + 1.08·16-s − 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $-0.674 - 0.738i$
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: \((2,\ 61,\ (\ :7/2),\ -0.674 - 0.738i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.644057 + 1.46105i\)
\(L(\frac12)\) \(\approx\) \(0.644057 + 1.46105i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (1.19e6 + 1.30e6i)T \)
good2 \( 1 + 19.6iT - 128T^{2} \)
3 \( 1 + 11.3T + 2.18e3T^{2} \)
5 \( 1 - 547.T + 7.81e4T^{2} \)
7 \( 1 + 1.20e3iT - 8.23e5T^{2} \)
11 \( 1 + 1.97e3iT - 1.94e7T^{2} \)
13 \( 1 - 3.52e3T + 6.27e7T^{2} \)
17 \( 1 + 2.13e4iT - 4.10e8T^{2} \)
19 \( 1 + 3.97e4T + 8.93e8T^{2} \)
23 \( 1 - 4.45e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.94e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.26e5iT - 2.75e10T^{2} \)
37 \( 1 + 5.66e4iT - 9.49e10T^{2} \)
41 \( 1 + 5.80e5T + 1.94e11T^{2} \)
43 \( 1 - 5.57e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.44e5T + 5.06e11T^{2} \)
53 \( 1 + 1.06e6iT - 1.17e12T^{2} \)
59 \( 1 + 4.39e4iT - 2.48e12T^{2} \)
67 \( 1 + 1.61e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.27e6iT - 9.09e12T^{2} \)
73 \( 1 + 4.52e5T + 1.10e13T^{2} \)
79 \( 1 + 1.82e5iT - 1.92e13T^{2} \)
83 \( 1 - 8.04e6T + 2.71e13T^{2} \)
89 \( 1 + 9.67e6iT - 4.42e13T^{2} \)
97 \( 1 - 4.82e6T + 8.07e13T^{2} \)
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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

−12.97456034717401900210201053496, −11.42110596998024112853202092295, −10.58105813501574886387679093937, −9.842588591796313770506766876314, −8.767067297803237727133546199533, −6.34414636142208508345065938004, −4.91507761219307116391657027667, −3.16117071314091417179817942314, −1.84140019593219818111044991655, −0.58945456269561840188095448707, 2.18090555674253863584563097872, 5.09931170611910557612457092112, 5.96850292317948557733314319267, 6.43605762441590171271177792064, 8.589091115721507794448689484051, 9.023896723851727320853586629010, 10.47750680655939977298369631926, 12.56179716099156206702896331197, 13.52299186728949537891804304931, 14.62188368461741801658399888288

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