↑

Properties

Label 2-187-17.4-c1-0-6
Degree $2$
Conductor $187$
Sign $0.998 + 0.0502i$
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  − 0.706i·2-s + (−1.00 + 1.00i)3-s + 1.50·4-s + (0.325 − 0.325i)5-s + (0.709 + 0.709i)6-s + (0.927 + 0.927i)7-s − 2.47i·8-s + 0.979i·9-s + (−0.229 − 0.229i)10-s + (0.707 + 0.707i)11-s + (−1.50 + 1.50i)12-s + 2.58·13-s + (0.654 − 0.654i)14-s + 0.654i·15-s + 1.25·16-s + (4.11 − 0.297i)17-s + ⋯
L(s)  = 1  − 0.499i·2-s + (−0.580 + 0.580i)3-s + 0.750·4-s + (0.145 − 0.145i)5-s + (0.289 + 0.289i)6-s + (0.350 + 0.350i)7-s − 0.874i·8-s + 0.326i·9-s + (−0.0726 − 0.0726i)10-s + (0.213 + 0.213i)11-s + (−0.435 + 0.435i)12-s + 0.717·13-s + (0.174 − 0.174i)14-s + 0.168i·15-s + 0.314·16-s + (0.997 − 0.0721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.998 + 0.0502i$
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: \((2,\ 187,\ (\ :1/2),\ 0.998 + 0.0502i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25538 - 0.0315625i\)
\(L(\frac12)\) \(\approx\) \(1.25538 - 0.0315625i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.11 + 0.297i)T \)
good2 \( 1 + 0.706iT - 2T^{2} \)
3 \( 1 + (1.00 - 1.00i)T - 3iT^{2} \)
5 \( 1 + (-0.325 + 0.325i)T - 5iT^{2} \)
7 \( 1 + (-0.927 - 0.927i)T + 7iT^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
19 \( 1 + 2.16iT - 19T^{2} \)
23 \( 1 + (6.56 + 6.56i)T + 23iT^{2} \)
29 \( 1 + (4.94 - 4.94i)T - 29iT^{2} \)
31 \( 1 + (1.80 - 1.80i)T - 31iT^{2} \)
37 \( 1 + (4.41 - 4.41i)T - 37iT^{2} \)
41 \( 1 + (3.32 + 3.32i)T + 41iT^{2} \)
43 \( 1 + 1.85iT - 43T^{2} \)
47 \( 1 + 1.58T + 47T^{2} \)
53 \( 1 + 6.95iT - 53T^{2} \)
59 \( 1 - 6.58iT - 59T^{2} \)
61 \( 1 + (-2.83 - 2.83i)T + 61iT^{2} \)
67 \( 1 + 16.2T + 67T^{2} \)
71 \( 1 + (1.41 - 1.41i)T - 71iT^{2} \)
73 \( 1 + (-8.10 + 8.10i)T - 73iT^{2} \)
79 \( 1 + (-6.62 - 6.62i)T + 79iT^{2} \)
83 \( 1 + 15.7iT - 83T^{2} \)
89 \( 1 + 2.28T + 89T^{2} \)
97 \( 1 + (-0.736 + 0.736i)T - 97iT^{2} \)
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

−12.22376275865354670288981838114, −11.54704809751725162002104010343, −10.65619976084603825669399158702, −10.03233542441898252093372310655, −8.692924332375134958366213325543, −7.38693925914822409206396896125, −6.09454500380338353302475931901, −5.07189174017693426574367699015, −3.58155123376373896239672794715, −1.86463957694462262648195702938, 1.62719029545517498795065065305, 3.66372743107663922637080975997, 5.71932783283176981382156172710, 6.18391635366754014572866444744, 7.38767620140626158850903276267, 8.103767236576348244441528332092, 9.699291179743173190606791015144, 10.87883416407405033436336629758, 11.68228472259803163997003097520, 12.34560706321104397486127379130

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