Properties

Label 2-187-11.3-c1-0-3
Degree $2$
Conductor $187$
Sign $0.927 - 0.374i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.77 + 1.28i)2-s + (−0.227 − 0.701i)3-s + (0.868 − 2.67i)4-s + (−1.17 − 0.855i)5-s + (1.30 + 0.951i)6-s + (−0.986 + 3.03i)7-s + (0.550 + 1.69i)8-s + (1.98 − 1.44i)9-s + 3.19·10-s + (3.04 − 1.31i)11-s − 2.07·12-s + (4.56 − 3.31i)13-s + (−2.16 − 6.66i)14-s + (−0.331 + 1.02i)15-s + (1.39 + 1.00i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−1.25 + 0.911i)2-s + (−0.131 − 0.405i)3-s + (0.434 − 1.33i)4-s + (−0.526 − 0.382i)5-s + (0.534 + 0.388i)6-s + (−0.372 + 1.14i)7-s + (0.194 + 0.598i)8-s + (0.662 − 0.481i)9-s + 1.00·10-s + (0.918 − 0.396i)11-s − 0.598·12-s + (1.26 − 0.919i)13-s + (−0.578 − 1.78i)14-s + (−0.0856 + 0.263i)15-s + (0.347 + 0.252i)16-s + (0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593288 + 0.115237i\)
\(L(\frac12)\) \(\approx\) \(0.593288 + 0.115237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.04 + 1.31i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (1.77 - 1.28i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.227 + 0.701i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.17 + 0.855i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.986 - 3.03i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.56 + 3.31i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (-2.23 - 6.86i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.04T + 23T^{2} \)
29 \( 1 + (-1.16 + 3.57i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.75 + 6.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.55 - 7.86i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.426 + 1.31i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.75T + 43T^{2} \)
47 \( 1 + (2.17 + 6.68i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.99 - 2.17i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.15 + 5.92i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.586T + 67T^{2} \)
71 \( 1 + (1.66 + 1.20i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.917 + 2.82i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.73 - 5.62i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.2 - 9.62i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (5.51 - 4.00i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.34714517022400816715609527334, −11.85910766721309687013420926968, −10.24638281039136351314660679698, −9.401534309264677302579588929202, −8.368647957792152468558290111050, −7.903538756946426785102739650349, −6.31433823607521593563031335668, −5.99235699952214755470630018908, −3.73070686166434267150191672902, −1.09635310128844823176953325476, 1.31123718880300132559909384341, 3.37504454938035191380108371659, 4.45399273309801652446671189282, 6.78241244713260964738701454132, 7.51292586132381936483077935845, 8.907529189928606428217241911796, 9.639416858989852046338326378419, 10.66809715067612291221536086281, 11.10461094967035415915743497773, 12.03227829811784367258510611912

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