↑

Properties

Label 2-76-4.3-c4-0-28
Degree $2$
Conductor $76$
Sign $-0.625 + 0.779i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 + 3.77i)2-s − 14.8i·3-s + (−12.4 − 10.0i)4-s + 13.5·5-s + (56.0 + 19.7i)6-s + 39.3i·7-s + (54.3 − 33.8i)8-s − 139.·9-s + (−18.0 + 51.2i)10-s − 218. i·11-s + (−148. + 185. i)12-s − 320.·13-s + (−148. − 52.2i)14-s − 201. i·15-s + (55.4 + 249. i)16-s − 187.·17-s + ⋯
L(s)  = 1  + (−0.331 + 0.943i)2-s − 1.65i·3-s + (−0.779 − 0.625i)4-s + 0.543·5-s + (1.55 + 0.547i)6-s + 0.803i·7-s + (0.849 − 0.528i)8-s − 1.72·9-s + (−0.180 + 0.512i)10-s − 1.80i·11-s + (−1.03 + 1.28i)12-s − 1.89·13-s + (−0.758 − 0.266i)14-s − 0.897i·15-s + (0.216 + 0.976i)16-s − 0.648·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.779i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.625 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.625 + 0.779i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.625 + 0.779i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.328591 - 0.684999i\)
\(L(\frac12)\) \(\approx\) \(0.328591 - 0.684999i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 3.77i)T \)
19 \( 1 + 82.8iT \)
good3 \( 1 + 14.8iT - 81T^{2} \)
5 \( 1 - 13.5T + 625T^{2} \)
7 \( 1 - 39.3iT - 2.40e3T^{2} \)
11 \( 1 + 218. iT - 1.46e4T^{2} \)
13 \( 1 + 320.T + 2.85e4T^{2} \)
17 \( 1 + 187.T + 8.35e4T^{2} \)
23 \( 1 - 302. iT - 2.79e5T^{2} \)
29 \( 1 - 178.T + 7.07e5T^{2} \)
31 \( 1 + 747. iT - 9.23e5T^{2} \)
37 \( 1 - 1.39e3T + 1.87e6T^{2} \)
41 \( 1 - 431.T + 2.82e6T^{2} \)
43 \( 1 + 2.94e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.06e3iT - 4.87e6T^{2} \)
53 \( 1 - 738.T + 7.89e6T^{2} \)
59 \( 1 + 6.04e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.14e3T + 1.38e7T^{2} \)
67 \( 1 + 2.47e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.34e3iT - 2.54e7T^{2} \)
73 \( 1 - 973.T + 2.83e7T^{2} \)
79 \( 1 - 4.62e3iT - 3.89e7T^{2} \)
83 \( 1 - 7.69e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.50e3T + 6.27e7T^{2} \)
97 \( 1 - 8.34e3T + 8.85e7T^{2} \)
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

−13.58059770075966723405007785137, −12.60144560358348979128027473494, −11.37521245529068180232861510522, −9.532119704938393633765568578319, −8.440929410474290527027211842558, −7.45653545195916326962468295460, −6.27986322100500689963767934872, −5.47873602299577202034068404326, −2.29334673075217257466991459898, −0.41479593682487533654968052723, 2.42414890774086979641092206192, 4.25924095950430620829000106696, 4.86459476394561642330871634505, 7.41842487568528844605844144368, 9.177568771773811299467544894850, 10.09322635259988438386798713554, 10.21525181923466026821606670711, 11.71864893920219141263432253091, 12.87158027147723459009626979278, 14.31399773079565868224300386861

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