Properties

Label 2-187-11.9-c1-0-5
Degree $2$
Conductor $187$
Sign $0.211 - 0.977i$
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  + (−0.135 + 0.418i)2-s + (−0.0312 + 0.0226i)3-s + (1.46 + 1.06i)4-s + (0.919 + 2.82i)5-s + (−0.00524 − 0.0161i)6-s + (−2.52 − 1.83i)7-s + (−1.35 + 0.983i)8-s + (−0.926 + 2.85i)9-s − 1.30·10-s + (1.79 − 2.78i)11-s − 0.0696·12-s + (0.515 − 1.58i)13-s + (1.10 − 0.806i)14-s + (−0.0928 − 0.0674i)15-s + (0.889 + 2.73i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.0960 + 0.295i)2-s + (−0.0180 + 0.0130i)3-s + (0.730 + 0.530i)4-s + (0.411 + 1.26i)5-s + (−0.00213 − 0.00658i)6-s + (−0.954 − 0.693i)7-s + (−0.478 + 0.347i)8-s + (−0.308 + 0.950i)9-s − 0.413·10-s + (0.542 − 0.840i)11-s − 0.0201·12-s + (0.143 − 0.440i)13-s + (0.296 − 0.215i)14-s + (−0.0239 − 0.0174i)15-s + (0.222 + 0.684i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973407 + 0.785013i\)
\(L(\frac12)\) \(\approx\) \(0.973407 + 0.785013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-1.79 + 2.78i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.135 - 0.418i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.0312 - 0.0226i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.919 - 2.82i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.52 + 1.83i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.515 + 1.58i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (-0.628 + 0.456i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.02T + 23T^{2} \)
29 \( 1 + (-2.59 - 1.88i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.93 + 9.04i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.99 + 3.62i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.08 + 3.69i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.04T + 43T^{2} \)
47 \( 1 + (-0.198 + 0.144i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.20 + 6.79i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.436 - 0.316i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.49 - 7.68i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.78T + 67T^{2} \)
71 \( 1 + (-0.708 - 2.17i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.82 + 4.96i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.59 - 11.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.91 - 8.96i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.84T + 89T^{2} \)
97 \( 1 + (-3.53 + 10.8i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.99069394728530053134412173238, −11.53832367213527406497119707153, −10.83327767989644354623372287441, −10.10907528131949282046181222589, −8.609221887149189900076292009690, −7.41842579665993080849732119319, −6.67747459657929699924941717705, −5.80586272210517899653983391282, −3.54427321933499604388557881479, −2.64461574278705094533587217274, 1.34778698929333266390614842023, 3.06738700947469894252902543527, 4.92278057272378249441840450783, 6.12317449752694602330345821970, 6.86563636961625862988858460909, 8.859828538909576390355428454255, 9.327238181618238831884679033594, 10.17859666579763391839552922615, 11.76527767601617483835731383632, 12.20809968445025452405807294735

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