↑

Properties

Label 2-187-17.4-c1-0-7
Degree $2$
Conductor $187$
Sign $0.347 - 0.937i$
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  + 2.33i·2-s + (1.01 − 1.01i)3-s − 3.45·4-s + (2.72 − 2.72i)5-s + (2.37 + 2.37i)6-s + (0.562 + 0.562i)7-s − 3.38i·8-s + 0.923i·9-s + (6.36 + 6.36i)10-s + (−0.707 − 0.707i)11-s + (−3.51 + 3.51i)12-s − 0.362·13-s + (−1.31 + 1.31i)14-s − 5.55i·15-s + 1.01·16-s + (−0.951 + 4.01i)17-s + ⋯
L(s)  = 1  + 1.65i·2-s + (0.588 − 0.588i)3-s − 1.72·4-s + (1.21 − 1.21i)5-s + (0.971 + 0.971i)6-s + (0.212 + 0.212i)7-s − 1.19i·8-s + 0.307i·9-s + (2.01 + 2.01i)10-s + (−0.213 − 0.213i)11-s + (−1.01 + 1.01i)12-s − 0.100·13-s + (−0.351 + 0.351i)14-s − 1.43i·15-s + 0.252·16-s + (−0.230 + 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.347 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22970 + 0.855240i\)
\(L(\frac12)\) \(\approx\) \(1.22970 + 0.855240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.951 - 4.01i)T \)
good2 \( 1 - 2.33iT - 2T^{2} \)
3 \( 1 + (-1.01 + 1.01i)T - 3iT^{2} \)
5 \( 1 + (-2.72 + 2.72i)T - 5iT^{2} \)
7 \( 1 + (-0.562 - 0.562i)T + 7iT^{2} \)
13 \( 1 + 0.362T + 13T^{2} \)
19 \( 1 + 1.20iT - 19T^{2} \)
23 \( 1 + (5.12 + 5.12i)T + 23iT^{2} \)
29 \( 1 + (3.64 - 3.64i)T - 29iT^{2} \)
31 \( 1 + (1.42 - 1.42i)T - 31iT^{2} \)
37 \( 1 + (5.32 - 5.32i)T - 37iT^{2} \)
41 \( 1 + (-6.92 - 6.92i)T + 41iT^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 - 5.49T + 47T^{2} \)
53 \( 1 - 6.59iT - 53T^{2} \)
59 \( 1 + 4.35iT - 59T^{2} \)
61 \( 1 + (6.98 + 6.98i)T + 61iT^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + (-5.10 + 5.10i)T - 71iT^{2} \)
73 \( 1 + (-0.463 + 0.463i)T - 73iT^{2} \)
79 \( 1 + (0.337 + 0.337i)T + 79iT^{2} \)
83 \( 1 - 5.30iT - 83T^{2} \)
89 \( 1 - 2.22T + 89T^{2} \)
97 \( 1 + (-11.0 + 11.0i)T - 97iT^{2} \)
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

−13.17824685375604104119745211642, −12.44758858673844508111557845926, −10.45777892475060060395580517758, −9.101177067563434204484723184052, −8.562225749059921620343447419147, −7.78577107150201294092597369760, −6.47029459425802246292933336453, −5.56886734058338472469872236400, −4.66922290482188862080436081330, −1.96914645750166227839622650674, 2.06424244881936692734635147000, 3.03036954134995176828349426363, 4.12356155669439087904512569996, 5.80681942695551113377928371267, 7.34746051466556743324128641719, 9.175260218748741451153456583757, 9.673125180518992005622347911622, 10.38864800306164565764395452055, 11.17384087398439362893358531637, 12.20104109148887495293308504528

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