Properties

Label 2-187-11.3-c1-0-14
Degree $2$
Conductor $187$
Sign $-0.394 + 0.918i$
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  + (1.80 − 1.31i)2-s + (−0.809 − 2.48i)3-s + (0.927 − 2.85i)4-s + (1.61 + 1.17i)5-s + (−4.73 − 3.44i)6-s + (−0.5 + 1.53i)7-s + (−0.690 − 2.12i)8-s + (−3.11 + 2.26i)9-s + 4.47·10-s + (−1.23 + 3.07i)11-s − 7.85·12-s + (−0.5 + 0.363i)13-s + (1.11 + 3.44i)14-s + (1.61 − 4.97i)15-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (1.27 − 0.929i)2-s + (−0.467 − 1.43i)3-s + (0.463 − 1.42i)4-s + (0.723 + 0.525i)5-s + (−1.93 − 1.40i)6-s + (−0.188 + 0.581i)7-s + (−0.244 − 0.751i)8-s + (−1.03 + 0.755i)9-s + 1.41·10-s + (−0.372 + 0.927i)11-s − 2.26·12-s + (−0.138 + 0.100i)13-s + (0.298 + 0.919i)14-s + (0.417 − 1.28i)15-s + (0.202 + 0.146i)16-s + (0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.394 + 0.918i$
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: \((2,\ 187,\ (\ :1/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10736 - 1.68079i\)
\(L(\frac12)\) \(\approx\) \(1.10736 - 1.68079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.23 - 3.07i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-1.80 + 1.31i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.809 + 2.48i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.5 - 1.53i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.5 - 0.363i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (1.61 + 4.97i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 + (-1.76 + 5.42i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2 - 1.45i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2 - 6.15i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.38 - 10.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 + (2.76 + 8.50i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.16 - 6.65i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.23 + 13.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.23 + 3.07i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (1.23 + 0.898i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.85 + 8.78i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.54 - 6.20i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.61 - 3.35i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (4.85 - 3.52i)T + (29.9 - 92.2i)T^{2} \)
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

−12.35788683367863206247164378438, −11.78477830283318200554688729198, −10.74918150174954255737677919299, −9.692457067752878927387821138186, −7.896405129844566490260414363222, −6.57248991350613759187906580815, −5.94300978934995928942665262725, −4.70436657882740417083427405519, −2.69673227750230445618789894744, −1.91891036566958514816456115725, 3.50842847506302592328034652341, 4.38884228502284554837322036563, 5.55564347229200479180294415659, 5.89254221262024890730657080009, 7.50793756582336497190414494125, 8.955164210184305362635593834380, 10.08345815291662001357877197296, 10.80991868388265130587499344605, 12.21678931208028670264386436181, 13.13592174582979061050916056079

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