↑

Properties

Label 2-187-11.5-c1-0-1
Degree $2$
Conductor $187$
Sign $0.972 - 0.232i$
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  + (−0.768 − 2.36i)2-s + (1.93 + 1.40i)3-s + (−3.38 + 2.46i)4-s + (−1.37 + 4.22i)5-s + (1.83 − 5.64i)6-s + (−2.07 + 1.50i)7-s + (4.40 + 3.20i)8-s + (0.832 + 2.56i)9-s + 11.0·10-s + (2.74 − 1.85i)11-s − 9.99·12-s + (0.0241 + 0.0741i)13-s + (5.16 + 3.75i)14-s + (−8.58 + 6.23i)15-s + (1.59 − 4.92i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.543 − 1.67i)2-s + (1.11 + 0.809i)3-s + (−1.69 + 1.23i)4-s + (−0.614 + 1.89i)5-s + (0.749 − 2.30i)6-s + (−0.784 + 0.570i)7-s + (1.55 + 1.13i)8-s + (0.277 + 0.854i)9-s + 3.49·10-s + (0.828 − 0.559i)11-s − 2.88·12-s + (0.00668 + 0.0205i)13-s + (1.38 + 1.00i)14-s + (−2.21 + 1.61i)15-s + (0.399 − 1.23i)16-s + (0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.972 - 0.232i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.972 - 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.958543 + 0.112872i\)
\(L(\frac12)\) \(\approx\) \(0.958543 + 0.112872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.74 + 1.85i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.768 + 2.36i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.93 - 1.40i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.37 - 4.22i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.07 - 1.50i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.0241 - 0.0741i)T + (-10.5 + 7.64i)T^{2} \)
19 \( 1 + (-1.59 - 1.16i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.92T + 23T^{2} \)
29 \( 1 + (-2.32 + 1.69i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.960 - 2.95i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.328 - 0.239i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.41 - 2.48i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.96T + 43T^{2} \)
47 \( 1 + (-4.99 - 3.62i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.52 - 4.68i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.43 + 6.13i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.74 + 5.35i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 + (0.577 - 1.77i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.35 + 4.61i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.20 + 6.79i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.91 - 5.87i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.57T + 89T^{2} \)
97 \( 1 + (-2.04 - 6.29i)T + (-78.4 + 57.0i)T^{2} \)
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

−12.19667727971326803614611384778, −11.38884431710685524643266431412, −10.58551485861079993976337012917, −9.775307647390064556906042789257, −9.101130881788346030702295227967, −8.047302807158642727073177310980, −6.57510886548446797895564267597, −3.97001118055747470184101633114, −3.18925486326701226770122666864, −2.69766462854902198106671673585, 1.01244417686165417861650310589, 4.00924883353413505079497243269, 5.25153566824941765830253802495, 6.80669469582851409778468539646, 7.50097627520852906284664245506, 8.435798671288941841182974529168, 8.979981243420778414651313697634, 9.716691047743551822380634287829, 12.03622297975601232810196232128, 13.08862853504251461645694352738

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