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Properties

Label 2-187-11.3-c1-0-6
Degree $2$
Conductor $187$
Sign $0.927 - 0.374i$
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  + (0.965 − 0.701i)2-s + (0.418 + 1.28i)3-s + (−0.177 + 0.547i)4-s + (−0.131 − 0.0953i)5-s + (1.30 + 0.951i)6-s + (0.0598 − 0.184i)7-s + (0.949 + 2.92i)8-s + (0.940 − 0.683i)9-s − 0.193·10-s + (3.04 − 1.31i)11-s − 0.780·12-s + (−3.25 + 2.36i)13-s + (−0.0713 − 0.219i)14-s + (0.0679 − 0.209i)15-s + (2.03 + 1.47i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.496i)2-s + (0.241 + 0.744i)3-s + (−0.0889 + 0.273i)4-s + (−0.0586 − 0.0426i)5-s + (0.534 + 0.388i)6-s + (0.0226 − 0.0695i)7-s + (0.335 + 1.03i)8-s + (0.313 − 0.227i)9-s − 0.0611·10-s + (0.918 − 0.396i)11-s − 0.225·12-s + (−0.903 + 0.656i)13-s + (−0.0190 − 0.0587i)14-s + (0.0175 − 0.0539i)15-s + (0.509 + 0.369i)16-s + (0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.927 - 0.374i$
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: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67727 + 0.325786i\)
\(L(\frac12)\) \(\approx\) \(1.67727 + 0.325786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.04 + 1.31i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.965 + 0.701i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.418 - 1.28i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.131 + 0.0953i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0598 + 0.184i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.25 - 2.36i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (1.80 + 5.55i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 9.42T + 23T^{2} \)
29 \( 1 + (-2.45 + 7.55i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.530 + 0.385i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.983 + 3.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.71 + 5.29i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.95T + 43T^{2} \)
47 \( 1 + (-3.86 - 11.8i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.43 - 3.22i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.10 - 12.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.62 + 1.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.26T + 67T^{2} \)
71 \( 1 + (-6.55 - 4.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.56 + 4.81i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.07 + 2.23i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.62 - 3.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.62T + 89T^{2} \)
97 \( 1 + (-8.18 + 5.94i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.37195142288443522525799345925, −11.94988394592704167292114175306, −10.81666840791564459163548528425, −9.693958201683989546818297016571, −8.837754586849832523938302847160, −7.61007949210617922409452854150, −6.15145883496346209797574898944, −4.40241285913152671277975203443, −4.09973668058138523619272629035, −2.50233172518530709789166891019, 1.71446039578737199900129114519, 3.82300156514752023183145739167, 5.09411631345531722000967641010, 6.27631684404322195589305415350, 7.20060076605318216070153437373, 8.119615015525918705127866129232, 9.686838729983134640816565940422, 10.37056160349707556643406532878, 12.09723265199141456758226884223, 12.57638761288685450997724578435

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