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Properties

Label 2-187-11.4-c1-0-5
Degree $2$
Conductor $187$
Sign $0.611 - 0.791i$
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  + (0.905 + 0.657i)2-s + (−0.440 + 1.35i)3-s + (−0.231 − 0.711i)4-s + (1.95 − 1.41i)5-s + (−1.28 + 0.936i)6-s + (1.26 + 3.90i)7-s + (0.950 − 2.92i)8-s + (0.785 + 0.570i)9-s + 2.69·10-s + (−2.08 + 2.57i)11-s + 1.06·12-s + (−3.08 − 2.24i)13-s + (−1.42 + 4.37i)14-s + (1.06 + 3.26i)15-s + (1.57 − 1.14i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.640 + 0.465i)2-s + (−0.254 + 0.782i)3-s + (−0.115 − 0.355i)4-s + (0.872 − 0.633i)5-s + (−0.526 + 0.382i)6-s + (0.479 + 1.47i)7-s + (0.335 − 1.03i)8-s + (0.261 + 0.190i)9-s + 0.852·10-s + (−0.629 + 0.776i)11-s + 0.307·12-s + (−0.856 − 0.622i)13-s + (−0.379 + 1.16i)14-s + (0.273 + 0.843i)15-s + (0.393 − 0.285i)16-s + (−0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46543 + 0.720022i\)
\(L(\frac12)\) \(\approx\) \(1.46543 + 0.720022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.08 - 2.57i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.905 - 0.657i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.440 - 1.35i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.95 + 1.41i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.26 - 3.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.08 + 2.24i)T + (4.01 + 12.3i)T^{2} \)
19 \( 1 + (-2.26 + 6.97i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.29T + 23T^{2} \)
29 \( 1 + (2.52 + 7.77i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.05 - 2.94i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.492 + 1.51i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.87 + 8.85i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.68T + 43T^{2} \)
47 \( 1 + (0.539 - 1.66i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.92 - 1.40i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.313 - 0.964i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (11.7 - 8.56i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 8.31T + 67T^{2} \)
71 \( 1 + (-3.32 + 2.41i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.99 - 12.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.35 + 0.982i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.15 + 6.65i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.75T + 89T^{2} \)
97 \( 1 + (-13.8 - 10.0i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.94696519054911945921055065862, −11.95919969599284609317625763139, −10.53899853209279025798665325845, −9.695897658219162126033600641280, −9.064616097897988361549325365803, −7.45465591834480483851432987791, −5.86090533760414966170335096972, −5.13597081472720979162366072213, −4.67703103372194261298340300781, −2.25029278815392206891487823815, 1.78329341218462396523071897336, 3.38732488184288028232566261201, 4.70844982952650787014729174609, 6.14330249428717088114441441004, 7.29167057477773316517706308349, 8.003213955913063156589634966551, 9.818326377407720236613523575761, 10.67026765276787619524984690977, 11.62578797572539501611682846566, 12.55681720586458292587961805367

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