Properties

Label 2-76-19.14-c4-0-5
Degree $2$
Conductor $76$
Sign $-0.977 + 0.211i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.57 − 1.15i)3-s + (15.2 + 12.7i)5-s + (−12.2 − 21.1i)7-s + (−34.2 − 12.4i)9-s + (29.7 − 51.5i)11-s + (−294. + 51.9i)13-s + (−85.3 − 101. i)15-s + (−410. + 149. i)17-s + (−296. − 205. i)19-s + (55.7 + 153. i)21-s + (206. − 172. i)23-s + (−39.8 − 226. i)25-s + (678. + 391. i)27-s + (213. − 587. i)29-s + (−287. + 165. i)31-s + ⋯
L(s)  = 1  + (−0.730 − 0.128i)3-s + (0.609 + 0.511i)5-s + (−0.249 − 0.431i)7-s + (−0.422 − 0.153i)9-s + (0.246 − 0.426i)11-s + (−1.74 + 0.307i)13-s + (−0.379 − 0.452i)15-s + (−1.41 + 0.516i)17-s + (−0.822 − 0.569i)19-s + (0.126 + 0.347i)21-s + (0.389 − 0.326i)23-s + (−0.0637 − 0.361i)25-s + (0.931 + 0.537i)27-s + (0.254 − 0.699i)29-s + (−0.298 + 0.172i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.977 + 0.211i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0179388 - 0.167700i\)
\(L(\frac12)\) \(\approx\) \(0.0179388 - 0.167700i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (296. + 205. i)T \)
good3 \( 1 + (6.57 + 1.15i)T + (76.1 + 27.7i)T^{2} \)
5 \( 1 + (-15.2 - 12.7i)T + (108. + 615. i)T^{2} \)
7 \( 1 + (12.2 + 21.1i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-29.7 + 51.5i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (294. - 51.9i)T + (2.68e4 - 9.76e3i)T^{2} \)
17 \( 1 + (410. - 149. i)T + (6.39e4 - 5.36e4i)T^{2} \)
23 \( 1 + (-206. + 172. i)T + (4.85e4 - 2.75e5i)T^{2} \)
29 \( 1 + (-213. + 587. i)T + (-5.41e5 - 4.54e5i)T^{2} \)
31 \( 1 + (287. - 165. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.58e3iT - 1.87e6T^{2} \)
41 \( 1 + (526. + 92.7i)T + (2.65e6 + 9.66e5i)T^{2} \)
43 \( 1 + (-627. - 526. i)T + (5.93e5 + 3.36e6i)T^{2} \)
47 \( 1 + (655. + 238. i)T + (3.73e6 + 3.13e6i)T^{2} \)
53 \( 1 + (-2.05e3 - 2.45e3i)T + (-1.37e6 + 7.77e6i)T^{2} \)
59 \( 1 + (64.7 + 177. i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (2.55e3 - 2.14e3i)T + (2.40e6 - 1.36e7i)T^{2} \)
67 \( 1 + (-2.38e3 + 6.54e3i)T + (-1.54e7 - 1.29e7i)T^{2} \)
71 \( 1 + (3.04e3 - 3.62e3i)T + (-4.41e6 - 2.50e7i)T^{2} \)
73 \( 1 + (235. - 1.33e3i)T + (-2.66e7 - 9.71e6i)T^{2} \)
79 \( 1 + (-2.33e3 - 411. i)T + (3.66e7 + 1.33e7i)T^{2} \)
83 \( 1 + (6.43e3 + 1.11e4i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (1.13e4 - 1.99e3i)T + (5.89e7 - 2.14e7i)T^{2} \)
97 \( 1 + (-4.01e3 - 1.10e4i)T + (-6.78e7 + 5.69e7i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.25728729182392110793937499869, −12.07542101891552514015743417822, −11.02448779386762733402394342145, −10.08761921981029776287164086825, −8.773515347694894056117914964055, −6.94473042121437830045626900090, −6.19629919445790557575739353488, −4.63180493500280984883204291508, −2.51278646306902581715571219042, −0.086130287553374518184919324940, 2.31726820433143338553302875635, 4.75066997603862356862932971091, 5.69270754001891423783145299002, 7.07301994049240701643583027923, 8.812750498179756961202720952222, 9.779451982462966086102104880938, 11.01003577731059287626516385250, 12.17909105479313628891515729658, 12.94029553762044113922515472440, 14.30231759270588186375814646984

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