↑

Properties

Label 2-187-11.9-c1-0-15
Degree $2$
Conductor $187$
Sign $-0.703 + 0.710i$
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  + (0.780 − 2.40i)2-s + (1.77 − 1.28i)3-s + (−3.54 − 2.57i)4-s + (0.678 + 2.08i)5-s + (−1.71 − 5.26i)6-s + (0.932 + 0.677i)7-s + (−4.85 + 3.52i)8-s + (0.556 − 1.71i)9-s + 5.54·10-s + (−3.01 + 1.38i)11-s − 9.59·12-s + (−0.288 + 0.887i)13-s + (2.35 − 1.71i)14-s + (3.89 + 2.82i)15-s + (1.97 + 6.09i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.551 − 1.69i)2-s + (1.02 − 0.743i)3-s + (−1.77 − 1.28i)4-s + (0.303 + 0.933i)5-s + (−0.698 − 2.14i)6-s + (0.352 + 0.256i)7-s + (−1.71 + 1.24i)8-s + (0.185 − 0.571i)9-s + 1.75·10-s + (−0.908 + 0.417i)11-s − 2.76·12-s + (−0.0799 + 0.246i)13-s + (0.629 − 0.457i)14-s + (1.00 + 0.729i)15-s + (0.494 + 1.52i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.703 + 0.710i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722513 - 1.73207i\)
\(L(\frac12)\) \(\approx\) \(0.722513 - 1.73207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (3.01 - 1.38i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.780 + 2.40i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.77 + 1.28i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.678 - 2.08i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.932 - 0.677i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.288 - 0.887i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (4.23 - 3.08i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 + (6.60 + 4.80i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.19 + 6.76i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.81 + 6.40i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.89 + 5.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.67T + 43T^{2} \)
47 \( 1 + (-2.54 + 1.84i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.15 - 9.71i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.17 - 3.75i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.628 + 1.93i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + (-4.48 - 13.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.700 - 0.508i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.06 - 3.28i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.31 - 4.03i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + (-0.177 + 0.547i)T + (-78.4 - 57.0i)T^{2} \)
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

−12.46716018740517729289858842417, −11.18334233495850166163580119742, −10.53603307061441206683541817810, −9.515400397835772168250716847154, −8.386449633795990853528005545772, −7.17922705233516523575415866427, −5.50228772333690363769786190344, −3.92066606265822806233176828376, −2.58394281781140736914346656772, −2.04937498765869809712094434428, 3.29300477794381158134689266422, 4.72985090407223239288870039379, 5.22371197667182485952687258371, 6.77660671952726149816256882302, 8.019930842578393130410052833869, 8.666480150897242378953022362351, 9.337104562494689690570737335611, 10.77059989672180157164682579968, 12.72629342417538786066814328009, 13.27108975489243047001364714001

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