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Properties

Label 2-187-11.3-c1-0-13
Degree $2$
Conductor $187$
Sign $-0.970 - 0.242i$
Analytic cond. $1.49320$
Root an. cond. $1.22196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.363i)2-s + (−0.690 − 2.12i)3-s + (−0.5 + 1.53i)4-s + (−1.80 − 1.31i)5-s + (1.11 + 0.812i)6-s + (−0.881 + 2.71i)7-s + (−0.690 − 2.12i)8-s + (−1.61 + 1.17i)9-s + 1.38·10-s + (−3.23 − 0.726i)11-s + 3.61·12-s + (−4.92 + 3.57i)13-s + (−0.545 − 1.67i)14-s + (−1.54 + 4.75i)15-s + (−1.49 − 1.08i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.256i)2-s + (−0.398 − 1.22i)3-s + (−0.250 + 0.769i)4-s + (−0.809 − 0.587i)5-s + (0.456 + 0.331i)6-s + (−0.333 + 1.02i)7-s + (−0.244 − 0.751i)8-s + (−0.539 + 0.391i)9-s + 0.437·10-s + (−0.975 − 0.219i)11-s + 1.04·12-s + (−1.36 + 0.992i)13-s + (−0.145 − 0.448i)14-s + (−0.398 + 1.22i)15-s + (−0.374 − 0.272i)16-s + (0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.970 - 0.242i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.23 + 0.726i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.5 - 0.363i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.690 + 2.12i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.80 + 1.31i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.881 - 2.71i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.92 - 3.57i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.76T + 23T^{2} \)
29 \( 1 + (-0.736 + 2.26i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.85 + 2.80i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.38 + 7.33i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.236 - 0.726i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (-0.690 - 2.12i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.881 + 0.640i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.5 - 10.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-10.1 - 7.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 + (6.35 + 4.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.718 + 2.21i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.16 + 5.93i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.09T + 89T^{2} \)
97 \( 1 + (12.3 - 8.95i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.04696415644978836203331396773, −11.80550047433774738134510227834, −9.805276323730996954625228927755, −8.695016609794056587601957608306, −7.85079571484695871372951839757, −7.17543296530014568714924071792, −5.90687347257439671492675061837, −4.40554090035931714098298634255, −2.53156401218874593706356067482, 0, 3.11573208045150655803109404960, 4.56019423578592024728610021389, 5.33272987153665917665333566760, 7.03810647883256020754070299075, 8.108134054109026181132709642568, 9.804976596276927917354167515250, 10.14758869818678459386592105767, 10.75833333809662181453262701440, 11.69625496364040515265837483627

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