↑

Properties

Label 2-76-76.75-c5-0-42
Degree $2$
Conductor $76$
Sign $0.989 - 0.144i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.48 + 1.37i)2-s + 24.7·3-s + (28.2 + 15.0i)4-s − 13.4·5-s + (135. + 33.9i)6-s − 215. i·7-s + (134. + 121. i)8-s + 369.·9-s + (−73.5 − 18.3i)10-s − 160. i·11-s + (699. + 372. i)12-s + 824. i·13-s + (295. − 1.18e3i)14-s − 331.·15-s + (571. + 849. i)16-s − 1.09e3·17-s + ⋯
L(s)  = 1  + (0.970 + 0.242i)2-s + 1.58·3-s + (0.882 + 0.470i)4-s − 0.239·5-s + (1.54 + 0.384i)6-s − 1.66i·7-s + (0.742 + 0.669i)8-s + 1.52·9-s + (−0.232 − 0.0580i)10-s − 0.400i·11-s + (1.40 + 0.746i)12-s + 1.35i·13-s + (0.402 − 1.61i)14-s − 0.380·15-s + (0.558 + 0.829i)16-s − 0.918·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.989 - 0.144i)\)

Particular Values

\(L(3)\) \(\approx\) \(4.88562 + 0.354736i\)
\(L(\frac12)\) \(\approx\) \(4.88562 + 0.354736i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.48 - 1.37i)T \)
19 \( 1 + (1.26e3 - 932. i)T \)
good3 \( 1 - 24.7T + 243T^{2} \)
5 \( 1 + 13.4T + 3.12e3T^{2} \)
7 \( 1 + 215. iT - 1.68e4T^{2} \)
11 \( 1 + 160. iT - 1.61e5T^{2} \)
13 \( 1 - 824. iT - 3.71e5T^{2} \)
17 \( 1 + 1.09e3T + 1.41e6T^{2} \)
23 \( 1 - 1.26e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.17e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.31e3T + 2.86e7T^{2} \)
37 \( 1 + 2.80e3iT - 6.93e7T^{2} \)
41 \( 1 + 3.92e3iT - 1.15e8T^{2} \)
43 \( 1 + 2.35e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.32e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.90e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.29e4T + 7.14e8T^{2} \)
61 \( 1 - 2.95e4T + 8.44e8T^{2} \)
67 \( 1 + 4.09e4T + 1.35e9T^{2} \)
71 \( 1 + 2.96e4T + 1.80e9T^{2} \)
73 \( 1 + 2.09e4T + 2.07e9T^{2} \)
79 \( 1 - 4.90e4T + 3.07e9T^{2} \)
83 \( 1 - 1.20e5iT - 3.93e9T^{2} \)
89 \( 1 + 3.71e4iT - 5.58e9T^{2} \)
97 \( 1 + 8.91e4iT - 8.58e9T^{2} \)
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

−13.87169846988682876703800299118, −13.05551167491995461178698903078, −11.49087663620704190161119702258, −10.24087347190306694457594321731, −8.685000081951451441910170394608, −7.59495103178741414583922542090, −6.70257291860729711207220139118, −4.27177760605368366320268340410, −3.67604028039030984760727103835, −1.96841812545924894043220814982, 2.23518429369802874869270011572, 2.95612134226744131631309090813, 4.56456420880136093742626455178, 6.16814762238912006123869895391, 7.85037792572520210356560459120, 8.828679988601214528015625811407, 10.07388572020367748755527439516, 11.64127631582089601782493438512, 12.75939768607573243476003720071, 13.40684898571976973096622556354

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