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Properties

Label 2-187-11.3-c1-0-2
Degree $2$
Conductor $187$
Sign $-0.195 - 0.980i$
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  + (−1.28 + 0.935i)2-s + (0.00348 + 0.0107i)3-s + (0.164 − 0.507i)4-s + (2.53 + 1.84i)5-s + (−0.0145 − 0.0105i)6-s + (−0.112 + 0.346i)7-s + (−0.721 − 2.22i)8-s + (2.42 − 1.76i)9-s − 4.99·10-s + (0.725 + 3.23i)11-s + 0.00602·12-s + (−4.04 + 2.93i)13-s + (−0.179 − 0.551i)14-s + (−0.0109 + 0.0337i)15-s + (3.86 + 2.81i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.910 + 0.661i)2-s + (0.00201 + 0.00619i)3-s + (0.0824 − 0.253i)4-s + (1.13 + 0.825i)5-s + (−0.00593 − 0.00431i)6-s + (−0.0425 + 0.131i)7-s + (−0.255 − 0.784i)8-s + (0.808 − 0.587i)9-s − 1.57·10-s + (0.218 + 0.975i)11-s + 0.00173·12-s + (−1.12 + 0.815i)13-s + (−0.0479 − 0.147i)14-s + (−0.00282 + 0.00870i)15-s + (0.967 + 0.702i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.195 - 0.980i$
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: \((2,\ 187,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539754 + 0.657870i\)
\(L(\frac12)\) \(\approx\) \(0.539754 + 0.657870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.725 - 3.23i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.28 - 0.935i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.00348 - 0.0107i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.53 - 1.84i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.112 - 0.346i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.04 - 2.93i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (-1.47 - 4.53i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.68T + 23T^{2} \)
29 \( 1 + (-2.90 + 8.93i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.32 - 4.59i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.53 + 7.79i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.686 + 2.11i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.31T + 43T^{2} \)
47 \( 1 + (1.25 + 3.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.1 + 7.36i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.12 + 3.45i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.65 + 3.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 8.99T + 67T^{2} \)
71 \( 1 + (-4.79 - 3.48i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.27 - 6.99i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.70 + 4.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.5 + 7.63i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.993T + 89T^{2} \)
97 \( 1 + (10.3 - 7.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.75600000424178806496384168298, −12.02163886481212343862447389367, −10.26429710251291001173860224050, −9.765421676922219676999067446227, −9.130500406520764533834422613311, −7.47298921687328421936141747591, −6.93046661000005318448532081930, −5.93353664957084897563110407945, −4.07916290956869198841426214184, −2.10453882924589320687491975383, 1.14899085174202185386086686474, 2.57463604647874104922371225002, 4.87619014267816327809505708259, 5.75930819469932977373771518281, 7.44783385089049757072679456170, 8.693723759906239495820255512848, 9.415282717179922361818353148864, 10.22955484388058890662099016386, 10.96260886209961969511201329958, 12.27076273488096880319046312505

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