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Properties

Label 2-187-17.16-c1-0-4
Degree $2$
Conductor $187$
Sign $-0.338 - 0.940i$
Analytic cond. $1.49320$
Root an. cond. $1.22196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.651·2-s + 2.48i·3-s − 1.57·4-s − 1.42i·5-s + 1.61i·6-s + 5.16i·7-s − 2.32·8-s − 3.17·9-s − 0.924i·10-s − i·11-s − 3.91i·12-s + 2.40·13-s + 3.36i·14-s + 3.52·15-s + 1.63·16-s + (−3.87 + 1.39i)17-s + ⋯
L(s)  = 1  + 0.460·2-s + 1.43i·3-s − 0.787·4-s − 0.635i·5-s + 0.660i·6-s + 1.95i·7-s − 0.823·8-s − 1.05·9-s − 0.292i·10-s − 0.301i·11-s − 1.13i·12-s + 0.668·13-s + 0.899i·14-s + 0.910·15-s + 0.408·16-s + (−0.940 + 0.338i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.338 - 0.940i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ -0.338 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.665330 + 0.946373i\)
\(L(\frac12)\) \(\approx\) \(0.665330 + 0.946373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + iT \)
17 \( 1 + (3.87 - 1.39i)T \)
good2 \( 1 - 0.651T + 2T^{2} \)
3 \( 1 - 2.48iT - 3T^{2} \)
5 \( 1 + 1.42iT - 5T^{2} \)
7 \( 1 - 5.16iT - 7T^{2} \)
13 \( 1 - 2.40T + 13T^{2} \)
19 \( 1 - 6.38T + 19T^{2} \)
23 \( 1 + 3.23iT - 23T^{2} \)
29 \( 1 - 1.63iT - 29T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 - 5.78iT - 37T^{2} \)
41 \( 1 + 9.96iT - 41T^{2} \)
43 \( 1 - 9.78T + 43T^{2} \)
47 \( 1 - 4.46T + 47T^{2} \)
53 \( 1 + 1.38T + 53T^{2} \)
59 \( 1 + 9.72T + 59T^{2} \)
61 \( 1 + 4.58iT - 61T^{2} \)
67 \( 1 + 7.82T + 67T^{2} \)
71 \( 1 - 8.20iT - 71T^{2} \)
73 \( 1 - 4.17iT - 73T^{2} \)
79 \( 1 - 3.00iT - 79T^{2} \)
83 \( 1 + 4.16T + 83T^{2} \)
89 \( 1 - 5.55T + 89T^{2} \)
97 \( 1 - 1.25iT - 97T^{2} \)
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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

−12.79724580242358238669715920263, −12.03051090820381501198761565551, −10.95391553375807712240881249761, −9.581221780207891839499811804760, −8.965958468046040441011110005287, −8.511858142124594286563082663371, −5.91258604167450240124475911859, −5.25200290764337763611169821284, −4.33484394436567182509666672304, −3.02847815534086221393718072960, 1.02674550437905237275154851910, 3.27060073962150401101631053377, 4.53414560447518519193020261348, 6.19109919700936786861842902077, 7.19157088062630225674781968843, 7.73682594958236681647984527544, 9.246226119951481049759694814718, 10.48824899296412936774492274216, 11.46382813007707154477620935692, 12.64512175659917601647364184517

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