↑

Properties

Label 2-234-117.110-c1-0-7
Degree $2$
Conductor $234$
Sign $0.874 - 0.484i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (−0.258 + 0.965i)2-s + (1.73 + 0.00498i)3-s + (−0.866 − 0.499i)4-s + (0.361 − 1.35i)5-s + (−0.453 + 1.67i)6-s + (−0.715 − 0.715i)7-s + (0.707 − 0.707i)8-s + (2.99 + 0.0172i)9-s + (1.21 + 0.699i)10-s + (5.61 + 1.50i)11-s + (−1.49 − 0.870i)12-s + (−0.218 + 3.59i)13-s + (0.876 − 0.505i)14-s + (0.633 − 2.33i)15-s + (0.500 + 0.866i)16-s + (−1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.999 + 0.00287i)3-s + (−0.433 − 0.249i)4-s + (0.161 − 0.603i)5-s + (−0.184 + 0.682i)6-s + (−0.270 − 0.270i)7-s + (0.249 − 0.249i)8-s + (0.999 + 0.00575i)9-s + (0.382 + 0.221i)10-s + (1.69 + 0.453i)11-s + (−0.432 − 0.251i)12-s + (−0.0606 + 0.998i)13-s + (0.234 − 0.135i)14-s + (0.163 − 0.603i)15-s + (0.125 + 0.216i)16-s + (−0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.874 - 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48559 + 0.383971i\)
\(L(\frac12)\) \(\approx\) \(1.48559 + 0.383971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-1.73 - 0.00498i)T \)
13 \( 1 + (0.218 - 3.59i)T \)
good5 \( 1 + (-0.361 + 1.35i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.715 + 0.715i)T + 7iT^{2} \)
11 \( 1 + (-5.61 - 1.50i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.07 + 1.89i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.580T + 23T^{2} \)
29 \( 1 + (0.892 - 0.515i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.52 + 2.28i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (7.82 - 2.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.15 - 1.15i)T + 41iT^{2} \)
43 \( 1 - 8.38iT - 43T^{2} \)
47 \( 1 + (0.388 + 1.45i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 2.77iT - 53T^{2} \)
59 \( 1 + (3.10 + 11.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + (0.802 - 0.802i)T - 67iT^{2} \)
71 \( 1 + (2.71 - 10.1i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.788 - 0.788i)T + 73iT^{2} \)
79 \( 1 + (0.827 - 1.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (15.7 - 4.21i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.783 - 2.92i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.70 + 8.70i)T - 97iT^{2} \)
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

−12.60605647427349043518487113859, −11.27612590264436316957825764429, −9.786762905548961121065863479862, −9.096384709656616514597186108496, −8.613098746648845220570519884187, −7.08911508935083289746505175869, −6.60992190698929842234354519452, −4.72615966995407827330159281353, −3.86660223023948450224631339518, −1.75921381894034822942112914357, 1.87285061981644706703732569124, 3.23827690610814825183028813746, 4.11112295337564417213465391192, 6.10019511013137214319282695210, 7.20613461967304763150689594377, 8.643974006315674513085362176038, 8.987101739807531915825708032602, 10.29477723473473475366689341039, 10.85230268152106656920112700343, 12.29535128930376558868813928854

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