↑

Properties

Label 2-187-11.4-c1-0-1
Degree $2$
Conductor $187$
Sign $0.456 - 0.889i$
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  + (−1.36 − 0.990i)2-s + (−0.936 + 2.88i)3-s + (0.260 + 0.801i)4-s + (2.28 − 1.66i)5-s + (4.13 − 3.00i)6-s + (0.207 + 0.638i)7-s + (−0.603 + 1.85i)8-s + (−5.00 − 3.63i)9-s − 4.76·10-s + (−1.58 + 2.91i)11-s − 2.55·12-s + (5.12 + 3.72i)13-s + (0.349 − 1.07i)14-s + (2.64 + 8.14i)15-s + (4.02 − 2.92i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.964 − 0.700i)2-s + (−0.540 + 1.66i)3-s + (0.130 + 0.400i)4-s + (1.02 − 0.742i)5-s + (1.68 − 1.22i)6-s + (0.0784 + 0.241i)7-s + (−0.213 + 0.656i)8-s + (−1.66 − 1.21i)9-s − 1.50·10-s + (−0.477 + 0.878i)11-s − 0.737·12-s + (1.42 + 1.03i)13-s + (0.0934 − 0.287i)14-s + (0.683 + 2.10i)15-s + (1.00 − 0.731i)16-s + (−0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548532 + 0.335120i\)
\(L(\frac12)\) \(\approx\) \(0.548532 + 0.335120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.58 - 2.91i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (1.36 + 0.990i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.936 - 2.88i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.28 + 1.66i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.207 - 0.638i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-5.12 - 3.72i)T + (4.01 + 12.3i)T^{2} \)
19 \( 1 + (2.20 - 6.77i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.166T + 23T^{2} \)
29 \( 1 + (0.320 + 0.986i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.424 + 0.308i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.48 - 4.58i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.90 + 5.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.938T + 43T^{2} \)
47 \( 1 + (-3.45 + 10.6i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.21 - 1.60i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.24 + 9.97i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.09 - 4.42i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + (6.35 - 4.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.33 + 4.10i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.56 - 1.13i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.785 - 0.570i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.27T + 89T^{2} \)
97 \( 1 + (6.24 + 4.53i)T + (29.9 + 92.2i)T^{2} \)
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

−12.32563051026748360107552530527, −11.35429691690581604722628536731, −10.44105217378706178441783296322, −9.918155642591569597655771852990, −9.114355283198413807475482716880, −8.498267506683309565705351117404, −6.02845234916761019106482211174, −5.24121701834052642901632113898, −3.99589141668415157635289424081, −1.86728672493959562740606258120, 0.889973647114564913316988496490, 2.78441512805131717180404571211, 5.83897833955696644489923275942, 6.30370372040915342086124971405, 7.24428451392809435637526241039, 8.116306321727301715095680143527, 9.048563994582718158879269929681, 10.63281676822628684876016469936, 11.11956963831863358960434369610, 12.76189915050494063364964259243

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