Properties

Label 2-76-19.12-c6-0-9
Degree $2$
Conductor $76$
Sign $-0.960 + 0.278i$
Analytic cond. $17.4841$
Root an. cond. $4.18140$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (36.3 − 20.9i)3-s + (−70.2 − 121. i)5-s − 229.·7-s + (515. − 892. i)9-s − 2.39e3·11-s + (−1.13e3 − 654. i)13-s + (−5.10e3 − 2.94e3i)15-s + (2.79e3 + 4.84e3i)17-s + (6.78e3 + 979. i)19-s + (−8.33e3 + 4.81e3i)21-s + (2.55e3 − 4.41e3i)23-s + (−2.04e3 + 3.54e3i)25-s − 1.26e4i·27-s + (−2.51e4 − 1.45e4i)29-s − 1.28e4i·31-s + ⋯
L(s)  = 1  + (1.34 − 0.776i)3-s + (−0.561 − 0.972i)5-s − 0.669·7-s + (0.706 − 1.22i)9-s − 1.79·11-s + (−0.515 − 0.297i)13-s + (−1.51 − 0.872i)15-s + (0.569 + 0.986i)17-s + (0.989 + 0.142i)19-s + (−0.900 + 0.519i)21-s + (0.209 − 0.363i)23-s + (−0.130 + 0.226i)25-s − 0.642i·27-s + (−1.03 − 0.596i)29-s − 0.429i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.960 + 0.278i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ -0.960 + 0.278i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.206692 - 1.45385i\)
\(L(\frac12)\) \(\approx\) \(0.206692 - 1.45385i\)
\(L(4)\) 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 \( 1 + (-6.78e3 - 979. i)T \)
good3 \( 1 + (-36.3 + 20.9i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (70.2 + 121. i)T + (-7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + 229.T + 1.17e5T^{2} \)
11 \( 1 + 2.39e3T + 1.77e6T^{2} \)
13 \( 1 + (1.13e3 + 654. i)T + (2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (-2.79e3 - 4.84e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
23 \( 1 + (-2.55e3 + 4.41e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.51e4 + 1.45e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + 1.28e4iT - 8.87e8T^{2} \)
37 \( 1 + 7.43e4iT - 2.56e9T^{2} \)
41 \( 1 + (-1.92e4 + 1.10e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (7.00e4 + 1.21e5i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-5.49e4 + 9.52e4i)T + (-5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-1.76e5 - 1.01e5i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (1.52e5 - 8.81e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-4.30e4 + 7.45e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-5.07e4 - 2.93e4i)T + (4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-3.69e5 + 2.13e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (1.00e5 + 1.74e5i)T + (-7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-3.53e5 + 2.04e5i)T + (1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 7.15e5T + 3.26e11T^{2} \)
89 \( 1 + (-7.25e5 - 4.18e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + (-1.03e6 + 5.95e5i)T + (4.16e11 - 7.21e11i)T^{2} \)
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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

−12.89771743072682380238021297227, −12.27211149529307120538135128123, −10.32086849264361721335288729908, −9.075641266266258741735450910220, −8.026159093024526050870073376814, −7.45674153057450414799228194425, −5.42756641947923484118981982207, −3.59253684398211763188684094198, −2.28213532720738458652423437280, −0.44290755579948686345639275123, 2.80846230082767765957914020098, 3.26617039079287596007115146581, 5.03255867676842191874301973017, 7.19362042388318088162840274442, 7.979739938048618921709675583506, 9.481964004516323667596850435184, 10.14958597487453156174242871948, 11.37433502477612493865237794228, 13.01202454932925514304686715489, 13.99229918661250810899816312657

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