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Properties

Label 2-187-11.4-c1-0-2
Degree $2$
Conductor $187$
Sign $0.659 - 0.751i$
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  + (−0.5 − 0.363i)2-s + (−0.344 + 1.05i)3-s + (−0.5 − 1.53i)4-s + (−0.901 + 0.654i)5-s + (0.557 − 0.404i)6-s + (1.34 + 4.13i)7-s + (−0.690 + 2.12i)8-s + (1.42 + 1.03i)9-s + 0.688·10-s + (2.24 − 2.44i)11-s + 1.80·12-s + (1.80 + 1.31i)13-s + (0.830 − 2.55i)14-s + (−0.383 − 1.18i)15-s + (−1.49 + 1.08i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.256i)2-s + (−0.198 + 0.611i)3-s + (−0.250 − 0.769i)4-s + (−0.403 + 0.292i)5-s + (0.227 − 0.165i)6-s + (0.508 + 1.56i)7-s + (−0.244 + 0.751i)8-s + (0.474 + 0.344i)9-s + 0.217·10-s + (0.677 − 0.735i)11-s + 0.520·12-s + (0.501 + 0.364i)13-s + (0.222 − 0.683i)14-s + (−0.0990 − 0.304i)15-s + (−0.374 + 0.272i)16-s + (0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.659 - 0.751i$
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: \((2,\ 187,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795791 + 0.360559i\)
\(L(\frac12)\) \(\approx\) \(0.795791 + 0.360559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.24 + 2.44i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (0.5 + 0.363i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.344 - 1.05i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.901 - 0.654i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.34 - 4.13i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.80 - 1.31i)T + (4.01 + 12.3i)T^{2} \)
19 \( 1 + (1.57 - 4.83i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.68T + 23T^{2} \)
29 \( 1 + (2.51 + 7.75i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.35 + 2.43i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.12 + 6.54i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.836 + 2.57i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.74T + 43T^{2} \)
47 \( 1 + (1.59 - 4.90i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.91 + 2.11i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.927 + 2.85i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.05 + 4.40i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.70T + 67T^{2} \)
71 \( 1 + (-12.9 + 9.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.79 + 5.52i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.4 - 8.29i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.201 - 0.146i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + (-7.65 - 5.56i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.45381589792705427449988457390, −11.26265504643596970303574845914, −11.02784985924513033475776418764, −9.608390359884102160690706085124, −9.022942145154713653371625039447, −7.923563481142975328098644212182, −6.07808810576007819526338870740, −5.35038323610511920920254050193, −3.93985581942489634653102862324, −1.94452181782367182720462367491, 1.03598939302934267626168419160, 3.72954212252259519017541071880, 4.57628089575792889903496855583, 6.81502949525641351912921107898, 7.20660685890291142585381249025, 8.190613721954177802628679673472, 9.274337226931716309674722602606, 10.50806720478910426163200719067, 11.60236935635707591469956812844, 12.63147659966590564736378397358

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