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Properties

Label 2-187-11.5-c1-0-9
Degree $2$
Conductor $187$
Sign $-0.773 + 0.634i$
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  + (−0.709 − 2.18i)2-s + (0.676 + 0.491i)3-s + (−2.65 + 1.92i)4-s + (0.327 − 1.00i)5-s + (0.593 − 1.82i)6-s + (2.54 − 1.85i)7-s + (2.37 + 1.72i)8-s + (−0.711 − 2.18i)9-s − 2.43·10-s + (−1.65 + 2.87i)11-s − 2.73·12-s + (−1.77 − 5.46i)13-s + (−5.85 − 4.25i)14-s + (0.716 − 0.520i)15-s + (0.0555 − 0.170i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.501 − 1.54i)2-s + (0.390 + 0.283i)3-s + (−1.32 + 0.962i)4-s + (0.146 − 0.450i)5-s + (0.242 − 0.745i)6-s + (0.962 − 0.699i)7-s + (0.838 + 0.609i)8-s + (−0.237 − 0.729i)9-s − 0.769·10-s + (−0.500 + 0.865i)11-s − 0.790·12-s + (−0.492 − 1.51i)13-s + (−1.56 − 1.13i)14-s + (0.185 − 0.134i)15-s + (0.0138 − 0.0427i)16-s + (0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ -0.773 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339016 - 0.947560i\)
\(L(\frac12)\) \(\approx\) \(0.339016 - 0.947560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (1.65 - 2.87i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.709 + 2.18i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.676 - 0.491i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.327 + 1.00i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.54 + 1.85i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.77 + 5.46i)T + (-10.5 + 7.64i)T^{2} \)
19 \( 1 + (-6.51 - 4.73i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.54T + 23T^{2} \)
29 \( 1 + (-0.207 + 0.150i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.441 - 1.35i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.15 - 1.56i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.91 - 5.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.16T + 43T^{2} \)
47 \( 1 + (4.92 + 3.57i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.95 - 6.00i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.27 + 3.10i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.200 - 0.615i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + (3.29 - 10.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.58 - 3.33i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.69 - 8.28i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.204 + 0.629i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + (0.574 + 1.76i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.17137807522680583408580759093, −11.13871735828094049423308511563, −10.03100883160837147609568885605, −9.741525805419735363289025985023, −8.384161429642494637339407869730, −7.61798806616127782464234213920, −5.34651586398795988368699665222, −4.05741729981848868881530227926, −2.83337978947635324139383719189, −1.15507533950654489884124163832, 2.42228427225014468086136465874, 4.86634044559895003793933711243, 5.77162465650688192822875171803, 7.00746074639437699222954040330, 7.84962269937346328435242108186, 8.617362118422527914766008199847, 9.464504580912011703352658558209, 10.96862749785279862813866480333, 11.88272129679696782347745068189, 13.62955487133920949635320429560

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