↑

Properties

Label 2-114-57.8-c1-0-7
Degree $2$
Conductor $114$
Sign $-0.740 + 0.671i$
Analytic cond. $0.910294$
Root an. cond. $0.954093$
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 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−3 − 1.73i)5-s + (−1.5 + 0.866i)6-s + 7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−3 + 1.73i)10-s − 3.46i·11-s + 1.73i·12-s + (4.5 − 2.59i)13-s + (0.5 − 0.866i)14-s + (3 + 5.19i)15-s + (−0.5 + 0.866i)16-s + (3 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−1.34 − 0.774i)5-s + (−0.612 + 0.353i)6-s + 0.377·7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.948 + 0.547i)10-s − 1.04i·11-s + 0.499i·12-s + (1.24 − 0.720i)13-s + (0.133 − 0.231i)14-s + (0.774 + 1.34i)15-s + (−0.125 + 0.216i)16-s + (0.727 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.740 + 0.671i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ -0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273905 - 0.709924i\)
\(L(\frac12)\) \(\approx\) \(0.273905 - 0.709924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
19 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 1.73i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 5.19iT - 37T^{2} \)
41 \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6 - 3.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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

−12.82243495678826302919942471214, −12.20904673297579849637342308621, −11.16667191009346429714088634197, −10.72444268687988517463598989558, −8.643800337145997528836872995992, −7.893468515568327552261057284843, −6.16234758361217626460643157504, −4.96684836277158435161736630940, −3.67396725005845432295325936298, −0.925932476813854479131803856221, 3.75486857896117614295779222679, 4.61031991713176461845988323493, 6.23502792569374287118793849084, 7.18684934220337001163323860335, 8.320897757636082964727315934810, 9.925093853600671066882103237684, 11.18408077480697387861441470150, 11.73925569930681308939593260603, 12.78951134301813824973529789799, 14.42523747140850661344326232074

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