Properties

Label 2-76-19.15-c4-0-0
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: $\chi_{76} (53, \cdot )$
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

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

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