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Properties

Label 2-187-11.9-c1-0-3
Degree $2$
Conductor $187$
Sign $0.993 - 0.115i$
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  + (0.648 − 1.99i)2-s + (−2.15 + 1.56i)3-s + (−1.94 − 1.41i)4-s + (1.10 + 3.41i)5-s + (1.72 + 5.31i)6-s + (0.243 + 0.177i)7-s + (−0.681 + 0.495i)8-s + (1.26 − 3.90i)9-s + 7.52·10-s + (3.21 + 0.801i)11-s + 6.40·12-s + (−1.93 + 5.95i)13-s + (0.511 − 0.371i)14-s + (−7.73 − 5.62i)15-s + (−0.938 − 2.88i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.458 − 1.41i)2-s + (−1.24 + 0.904i)3-s + (−0.971 − 0.705i)4-s + (0.495 + 1.52i)5-s + (0.705 + 2.17i)6-s + (0.0921 + 0.0669i)7-s + (−0.240 + 0.175i)8-s + (0.422 − 1.30i)9-s + 2.38·10-s + (0.970 + 0.241i)11-s + 1.84·12-s + (−0.536 + 1.65i)13-s + (0.136 − 0.0993i)14-s + (−1.99 − 1.45i)15-s + (−0.234 − 0.721i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.993 - 0.115i$
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: \((2,\ 187,\ (\ :1/2),\ 0.993 - 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08356 + 0.0628545i\)
\(L(\frac12)\) \(\approx\) \(1.08356 + 0.0628545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.21 - 0.801i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.648 + 1.99i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.15 - 1.56i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.10 - 3.41i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.243 - 0.177i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.93 - 5.95i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (-1.61 + 1.17i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + (2.05 + 1.49i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.14 + 6.58i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.01 - 5.82i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.24 - 2.35i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 + (-4.51 + 3.28i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.10 + 12.6i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.89 + 4.28i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.823 - 2.53i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.815T + 67T^{2} \)
71 \( 1 + (2.28 + 7.02i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.66 - 4.11i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.496 - 1.52i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.82 + 5.61i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 2.19T + 89T^{2} \)
97 \( 1 + (2.10 - 6.48i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.89175593601516486329573154187, −11.58096347740778618018213196005, −10.93827312376632190552853956558, −9.842982413964937214995317518952, −9.646561063755944988455295221055, −6.98345098444630188822632489068, −6.11633552424905887769551493723, −4.64221843082463742852286116091, −3.74752850094704812718901798380, −2.16471255059720946242355186650, 1.10607141263579417918270611993, 4.57435620867023243143224772183, 5.58714928492305822192211815211, 5.91237219374225325290419129470, 7.22257287102429886624647325864, 8.073819555725979626026563965518, 9.235147892274646924128849447315, 10.76667050452024088975774865301, 12.17880786908517494494197087226, 12.60750508500301117509409642319

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