↑

Properties

Label 2-187-11.3-c1-0-9
Degree $2$
Conductor $187$
Sign $0.860 + 0.509i$
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  + (−1.73 + 1.25i)2-s + (0.331 + 1.02i)3-s + (0.799 − 2.45i)4-s + (−2.02 − 1.47i)5-s + (−1.86 − 1.35i)6-s + (1.12 − 3.44i)7-s + (0.387 + 1.19i)8-s + (1.49 − 1.08i)9-s + 5.37·10-s + (−2.89 − 1.62i)11-s + 2.77·12-s + (−1.40 + 1.02i)13-s + (2.39 + 7.38i)14-s + (0.832 − 2.56i)15-s + (2.01 + 1.46i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−1.22 + 0.890i)2-s + (0.191 + 0.589i)3-s + (0.399 − 1.22i)4-s + (−0.907 − 0.659i)5-s + (−0.759 − 0.551i)6-s + (0.423 − 1.30i)7-s + (0.137 + 0.421i)8-s + (0.497 − 0.361i)9-s + 1.69·10-s + (−0.872 − 0.488i)11-s + 0.801·12-s + (−0.390 + 0.283i)13-s + (0.641 + 1.97i)14-s + (0.215 − 0.661i)15-s + (0.502 + 0.365i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.860 + 0.509i$
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: \((2,\ 187,\ (\ :1/2),\ 0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468464 - 0.128242i\)
\(L(\frac12)\) \(\approx\) \(0.468464 - 0.128242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.89 + 1.62i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.73 - 1.25i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.331 - 1.02i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.02 + 1.47i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.12 + 3.44i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.40 - 1.02i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (0.663 + 2.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 + (-0.564 + 1.73i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.18 + 2.31i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.735 + 2.26i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.48 + 10.7i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-4.03 - 12.4i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.38 - 3.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.42 + 13.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.99 - 4.35i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.79T + 67T^{2} \)
71 \( 1 + (-6.61 - 4.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.32 + 4.06i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.91 + 2.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.8 - 7.86i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.84T + 89T^{2} \)
97 \( 1 + (4.19 - 3.04i)T + (29.9 - 92.2i)T^{2} \)
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

−12.47529991302422549641522870547, −11.07332559441067487714654622081, −10.27112477603998217144733141662, −9.339355374344619520433824112071, −8.333255471598095977792222945440, −7.63266221734303793552444103623, −6.75283630606410478001891741803, −4.88056116141656616616357901293, −3.84973932517541626638381386494, −0.64318625907524384783624581913, 1.94956729446933533862865609148, 2.98320673523311113993598013044, 5.08830255618499309896298108580, 6.97627637891634290951010180886, 8.004725111644260095557927298655, 8.443503199348686952866365660171, 9.862421410476375997305472870851, 10.63687220526861962834540175269, 11.67688837055888333931113500507, 12.21231035305173246436388046471

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