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Properties

Label 2-187-11.9-c1-0-10
Degree $2$
Conductor $187$
Sign $0.944 + 0.329i$
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  + (−0.0911 + 0.280i)2-s + (1.69 − 1.23i)3-s + (1.54 + 1.12i)4-s + (−0.738 − 2.27i)5-s + (0.190 + 0.587i)6-s + (0.570 + 0.414i)7-s + (−0.933 + 0.678i)8-s + (0.429 − 1.32i)9-s + 0.705·10-s + (−2.54 − 2.12i)11-s + 4.00·12-s + (−0.851 + 2.62i)13-s + (−0.168 + 0.122i)14-s + (−4.05 − 2.94i)15-s + (1.07 + 3.31i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.0644 + 0.198i)2-s + (0.978 − 0.711i)3-s + (0.773 + 0.562i)4-s + (−0.330 − 1.01i)5-s + (0.0779 + 0.239i)6-s + (0.215 + 0.156i)7-s + (−0.330 + 0.239i)8-s + (0.143 − 0.440i)9-s + 0.222·10-s + (−0.767 − 0.641i)11-s + 1.15·12-s + (−0.236 + 0.726i)13-s + (−0.0449 + 0.0326i)14-s + (−1.04 − 0.760i)15-s + (0.269 + 0.828i)16-s + (−0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.944 + 0.329i$
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: \((2,\ 187,\ (\ :1/2),\ 0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57995 - 0.268076i\)
\(L(\frac12)\) \(\approx\) \(1.57995 - 0.268076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.54 + 2.12i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.0911 - 0.280i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.69 + 1.23i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.738 + 2.27i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.570 - 0.414i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.851 - 2.62i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (3.29 - 2.39i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.38T + 23T^{2} \)
29 \( 1 + (-2.77 - 2.01i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.346 + 1.06i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.07 + 4.41i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.50 - 4.72i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.01T + 43T^{2} \)
47 \( 1 + (5.17 - 3.76i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.290 + 0.893i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.0781 + 0.0567i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.993 + 3.05i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.44T + 67T^{2} \)
71 \( 1 + (4.43 + 13.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.54 - 6.93i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.00 + 15.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.735 - 2.26i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + (3.80 - 11.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.65140339235147540765725156401, −11.81924594037545429441797979263, −10.72606246018406861828402591554, −8.955182600905423260907193044896, −8.377843712667460723668085822372, −7.67267562767967251074037637755, −6.57911700028371745233774199827, −4.96828775305094729165418315310, −3.25213666082050406454401537822, −1.94168929167169809105559901415, 2.46097480371085671956976166277, 3.32336578078363823701008892701, 4.94859926246064495468579477534, 6.58825661984882681718808551485, 7.50710831975170153989983766438, 8.683755352615747855771630767649, 10.04021633038040664815977733409, 10.44807630044298825467461454836, 11.33282685912976766776985612671, 12.58654087963935697222131307175

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