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Properties

Label 2-187-11.9-c1-0-14
Degree $2$
Conductor $187$
Sign $-0.836 + 0.548i$
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  + (0.690 − 2.12i)2-s + (0.309 − 0.224i)3-s + (−2.42 − 1.76i)4-s + (−0.618 − 1.90i)5-s + (−0.263 − 0.812i)6-s + (−0.5 − 0.363i)7-s + (−1.80 + 1.31i)8-s + (−0.881 + 2.71i)9-s − 4.47·10-s + (3.23 − 0.726i)11-s − 1.14·12-s + (−0.5 + 1.53i)13-s + (−1.11 + 0.812i)14-s + (−0.618 − 0.449i)15-s + (−0.309 − 0.951i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.488 − 1.50i)2-s + (0.178 − 0.129i)3-s + (−1.21 − 0.881i)4-s + (−0.276 − 0.850i)5-s + (−0.107 − 0.331i)6-s + (−0.188 − 0.137i)7-s + (−0.639 + 0.464i)8-s + (−0.293 + 0.904i)9-s − 1.41·10-s + (0.975 − 0.219i)11-s − 0.330·12-s + (−0.138 + 0.426i)13-s + (−0.298 + 0.217i)14-s + (−0.159 − 0.115i)15-s + (−0.0772 − 0.237i)16-s + (−0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.836 + 0.548i$
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: \((2,\ 187,\ (\ :1/2),\ -0.836 + 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413411 - 1.38342i\)
\(L(\frac12)\) \(\approx\) \(0.413411 - 1.38342i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.690 + 2.12i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.309 + 0.224i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.618 + 1.90i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.5 + 0.363i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.5 - 1.53i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (-0.618 + 0.449i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + (-6.23 - 4.53i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2 - 6.15i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2 + 1.45i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.61 + 4.08i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 + (7.23 - 5.25i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.33 - 4.11i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.236 + 0.171i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.236 - 0.726i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (-3.23 - 9.95i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.85 + 2.80i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.95 - 9.09i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.38 - 7.33i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9.79T + 89T^{2} \)
97 \( 1 + (-1.85 + 5.70i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.24435588270685315949576578211, −11.33937033039091065349011003558, −10.54061605458481706411006580646, −9.312648050021047493849436453192, −8.552955802524661452293086632971, −7.01004034752355615367760905988, −5.16939340926042099365614284823, −4.28447776786876204157893529551, −2.93808581829960457264585904877, −1.34804971277965267158476515704, 3.24606472589446669182366532576, 4.43530283476735211814027090786, 5.99360400812990818968601992516, 6.62694055045862284062426473793, 7.58723588781873925583788131206, 8.699522960340689816956276191657, 9.719262971516819509253384209392, 11.13477542192122288668095444531, 12.20398566633821186723800686029, 13.32350514687438351098370150873

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