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Properties

Label 2-76-19.11-c5-0-8
Degree $2$
Conductor $76$
Sign $-0.963 - 0.266i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.4 − 21.5i)3-s + (−18.0 + 31.3i)5-s − 208.·7-s + (−187. − 324. i)9-s − 204.·11-s + (−146. − 254. i)13-s + (449. + 779. i)15-s + (−850. + 1.47e3i)17-s + (1.32e3 − 841. i)19-s + (−2.59e3 + 4.49e3i)21-s + (−1.76e3 − 3.06e3i)23-s + (907. + 1.57e3i)25-s − 3.28e3·27-s + (−3.20e3 − 5.55e3i)29-s − 2.73e3·31-s + ⋯
L(s)  = 1  + (0.797 − 1.38i)3-s + (−0.323 + 0.560i)5-s − 1.61·7-s + (−0.771 − 1.33i)9-s − 0.509·11-s + (−0.240 − 0.417i)13-s + (0.516 + 0.894i)15-s + (−0.713 + 1.23i)17-s + (0.844 − 0.535i)19-s + (−1.28 + 2.22i)21-s + (−0.696 − 1.20i)23-s + (0.290 + 0.503i)25-s − 0.867·27-s + (−0.707 − 1.22i)29-s − 0.510·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.963 - 0.266i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ -0.963 - 0.266i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0788556 + 0.581090i\)
\(L(\frac12)\) \(\approx\) \(0.0788556 + 0.581090i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-1.32e3 + 841. i)T \)
good3 \( 1 + (-12.4 + 21.5i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (18.0 - 31.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + 208.T + 1.68e4T^{2} \)
11 \( 1 + 204.T + 1.61e5T^{2} \)
13 \( 1 + (146. + 254. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (850. - 1.47e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (1.76e3 + 3.06e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (3.20e3 + 5.55e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + 2.73e3T + 2.86e7T^{2} \)
37 \( 1 + 5.42e3T + 6.93e7T^{2} \)
41 \( 1 + (3.69e3 - 6.39e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-7.87e3 + 1.36e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-9.72e3 - 1.68e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (4.29e3 + 7.43e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.39e4 + 2.41e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.53e4 + 2.66e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.41e4 + 4.17e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (3.43e3 - 5.94e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-1.93e4 + 3.34e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.21e4 - 5.56e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.15e4T + 3.93e9T^{2} \)
89 \( 1 + (2.32e4 + 4.02e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (4.52e4 - 7.82e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.97104685427929839776012025674, −12.32696916400245748897872368287, −10.69104464965867374328872855145, −9.351416392215632022692248735371, −8.057230487962245472254946131750, −7.04854574135477244131186923573, −6.17510912314135763784207308814, −3.45445350777997280348705054280, −2.36607907462635469202430026700, −0.21359999621564018430305160857, 2.92951950227876081206348623703, 3.98246641707026522353827599385, 5.36542563495700807546019674217, 7.27861840487672649759671560151, 8.908515645563348296905583394179, 9.496000018983365936392952172643, 10.39884204955236904345108606822, 11.94287212699551076290180115433, 13.21340210762848228781143655722, 14.15566231351279592507345710521

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