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Properties

Label 2-187-11.5-c1-0-15
Degree $2$
Conductor $187$
Sign $0.354 - 0.935i$
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  + (−0.841 − 2.58i)2-s + (−1.62 − 1.18i)3-s + (−4.37 + 3.18i)4-s + (0.802 − 2.47i)5-s + (−1.69 + 5.20i)6-s + (−2.56 + 1.86i)7-s + (7.51 + 5.46i)8-s + (0.321 + 0.990i)9-s − 7.07·10-s + (0.00770 − 3.31i)11-s + 10.8·12-s + (0.911 + 2.80i)13-s + (6.99 + 5.08i)14-s + (−4.22 + 3.06i)15-s + (4.47 − 13.7i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.594 − 1.83i)2-s + (−0.939 − 0.682i)3-s + (−2.18 + 1.59i)4-s + (0.359 − 1.10i)5-s + (−0.690 + 2.12i)6-s + (−0.971 + 0.705i)7-s + (2.65 + 1.93i)8-s + (0.107 + 0.330i)9-s − 2.23·10-s + (0.00232 − 0.999i)11-s + 3.14·12-s + (0.252 + 0.778i)13-s + (1.86 + 1.35i)14-s + (−1.09 + 0.792i)15-s + (1.11 − 3.44i)16-s + (0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.354 - 0.935i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.354 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211276 + 0.145872i\)
\(L(\frac12)\) \(\approx\) \(0.211276 + 0.145872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.00770 + 3.31i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.841 + 2.58i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.62 + 1.18i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.802 + 2.47i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.56 - 1.86i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.911 - 2.80i)T + (-10.5 + 7.64i)T^{2} \)
19 \( 1 + (1.02 + 0.747i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.60T + 23T^{2} \)
29 \( 1 + (-0.790 + 0.574i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.895 + 2.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.29 - 5.29i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.38 + 1.73i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.98T + 43T^{2} \)
47 \( 1 + (7.61 + 5.53i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.46 + 7.58i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.79 - 4.21i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.32 + 10.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (-2.83 + 8.73i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.73 + 4.89i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.32 + 7.16i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.151 - 0.466i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 + (3.91 + 12.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.91012505720556877694552574449, −11.09835292239431331317834309800, −9.829012441547535710390764139951, −9.085645885890730734761041971784, −8.305505167411587405052036734088, −6.33308525466818169245612808244, −5.11789383373280936279945112814, −3.50976136754096878913931847499, −1.80944798256666156379446057412, −0.31403476666937658708832002496, 4.03773973512853843927512783393, 5.37498126635257531420227575898, 6.30034423925076891956972798440, 6.93967969028928068539749159526, 8.007414183275726909410725494761, 9.607243712813683305355222678671, 10.21791501695199490554067036922, 10.68171845716249342407989455007, 12.66951846082940529430147898890, 13.84933485354501265675672700046

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