↑

Properties

Label 2-187-11.3-c1-0-4
Degree $2$
Conductor $187$
Sign $0.678 - 0.734i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.363i)2-s + (0.118 + 0.363i)3-s + (−0.5 + 1.53i)4-s + (0.309 + 0.224i)5-s + (0.190 + 0.138i)6-s + (−0.881 + 2.71i)7-s + (0.690 + 2.12i)8-s + (2.30 − 1.67i)9-s + 0.236·10-s + (−2.19 + 2.48i)11-s − 0.618·12-s + (4.42 − 3.21i)13-s + (0.545 + 1.67i)14-s + (−0.0450 + 0.138i)15-s + (−1.49 − 1.08i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.256i)2-s + (0.0681 + 0.209i)3-s + (−0.250 + 0.769i)4-s + (0.138 + 0.100i)5-s + (0.0779 + 0.0566i)6-s + (−0.333 + 1.02i)7-s + (0.244 + 0.751i)8-s + (0.769 − 0.559i)9-s + 0.0746·10-s + (−0.660 + 0.750i)11-s − 0.178·12-s + (1.22 − 0.892i)13-s + (0.145 + 0.448i)14-s + (−0.0116 + 0.0358i)15-s + (−0.374 − 0.272i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.678 - 0.734i$
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: \((2,\ 187,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25932 + 0.551385i\)
\(L(\frac12)\) \(\approx\) \(1.25932 + 0.551385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.19 - 2.48i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.5 + 0.363i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.118 - 0.363i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.309 - 0.224i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.881 - 2.71i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.42 + 3.21i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.76T + 23T^{2} \)
29 \( 1 + (0.163 - 0.502i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2 + 1.45i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.73 + 8.42i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.04 - 3.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (1.04 + 3.21i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.09 - 6.60i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.92 + 9.00i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.73 - 1.26i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + (6.35 + 4.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.35 - 13.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.47 + 6.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.11 + 1.53i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.94T + 89T^{2} \)
97 \( 1 + (-5.23 + 3.80i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.81124761655411981478646417003, −11.99934606834451747652109897418, −10.84002839279729230715922567405, −9.697527407632561005936227168323, −8.734228295606415221446005298967, −7.74194113791909969857892394844, −6.34779590965563057510086277823, −5.02991586615380695657959641984, −3.75278852587815134239947882183, −2.53340850071694990803462456398, 1.36004752399487996331585650921, 3.76250013180098342928162978984, 4.89000595261024029391580181215, 6.19916084002690745632278404101, 7.05354659934358248328374574742, 8.347410884207180547560859118413, 9.633850022044551873589393628937, 10.50242615611347052756949520704, 11.24244943263492019388017280862, 13.03267872611378297088279514718

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