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Properties

Label 2-187-11.9-c1-0-12
Degree $2$
Conductor $187$
Sign $-0.909 + 0.414i$
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  + (0.5 − 1.53i)2-s + (−2.11 + 1.53i)3-s + (−0.5 − 0.363i)4-s + (−0.809 − 2.48i)5-s + (1.30 + 4.02i)6-s + (−3.11 − 2.26i)7-s + (1.80 − 1.31i)8-s + (1.19 − 3.66i)9-s − 4.23·10-s + (−3.30 + 0.224i)11-s + 1.61·12-s + (1.07 − 3.30i)13-s + (−5.04 + 3.66i)14-s + (5.54 + 4.02i)15-s + (−1.50 − 4.61i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.353 − 1.08i)2-s + (−1.22 + 0.888i)3-s + (−0.250 − 0.181i)4-s + (−0.361 − 1.11i)5-s + (0.534 + 1.64i)6-s + (−1.17 − 0.856i)7-s + (0.639 − 0.464i)8-s + (0.396 − 1.22i)9-s − 1.33·10-s + (−0.997 + 0.0676i)11-s + 0.467·12-s + (0.297 − 0.915i)13-s + (−1.34 + 0.979i)14-s + (1.43 + 1.04i)15-s + (−0.375 − 1.15i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.909 + 0.414i$
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: \((2,\ 187,\ (\ :1/2),\ -0.909 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138994 - 0.640080i\)
\(L(\frac12)\) \(\approx\) \(0.138994 - 0.640080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.30 - 0.224i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.5 + 1.53i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.11 - 1.53i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.809 + 2.48i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.11 + 2.26i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.07 + 3.30i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 + (-7.66 - 5.56i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2 + 6.15i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.73 + 1.26i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.54 - 3.30i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-4.54 + 3.30i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.09 + 6.43i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.427 + 0.310i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.73 + 8.42i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 7.14T + 67T^{2} \)
71 \( 1 + (-0.354 - 1.08i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.35 - 1.71i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.527 + 1.62i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.118 - 0.363i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.94T + 89T^{2} \)
97 \( 1 + (-0.763 + 2.35i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.26681218287248715987100757882, −11.03795295920387618684028183618, −10.40631125729659579993078788355, −9.884134076041464752241711033528, −8.265699344361634609416883914521, −6.68600051072726623529321846197, −5.26064434088219342548513394683, −4.38173317428995131135691486095, −3.32537605501136617936121604472, −0.59269947569777861929039822477, 2.68686174416927891712397322893, 4.94581206480141544099499221947, 6.09798733595973756524153543415, 6.68632724790809639084957490901, 7.16728463410902221703983914354, 8.636995788372200252791474807512, 10.41825617066262738832992651896, 11.14716081926067464753410300661, 12.10467933962075078997853296924, 13.07542053092833135166176083881

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