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Properties

Label 2-76-19.7-c5-0-4
Degree $2$
Conductor $76$
Sign $0.566 + 0.823i$
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  + (−14.7 − 25.5i)3-s + (35.6 + 61.7i)5-s + 252.·7-s + (−315. + 545. i)9-s + 88.0·11-s + (307. − 533. i)13-s + (1.05e3 − 1.82e3i)15-s + (285. + 494. i)17-s + (361. − 1.53e3i)19-s + (−3.72e3 − 6.45e3i)21-s + (−1.21e3 + 2.10e3i)23-s + (−977. + 1.69e3i)25-s + 1.14e4·27-s + (1.14e3 − 1.97e3i)29-s + 3.68e3·31-s + ⋯
L(s)  = 1  + (−0.947 − 1.64i)3-s + (0.637 + 1.10i)5-s + 1.94·7-s + (−1.29 + 2.24i)9-s + 0.219·11-s + (0.505 − 0.875i)13-s + (1.20 − 2.09i)15-s + (0.239 + 0.415i)17-s + (0.229 − 0.973i)19-s + (−1.84 − 3.19i)21-s + (−0.478 + 0.829i)23-s + (−0.312 + 0.542i)25-s + 3.02·27-s + (0.252 − 0.437i)29-s + 0.688·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.823i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.566 + 0.823i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.566 + 0.823i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.56074 - 0.820764i\)
\(L(\frac12)\) \(\approx\) \(1.56074 - 0.820764i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-361. + 1.53e3i)T \)
good3 \( 1 + (14.7 + 25.5i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-35.6 - 61.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 252.T + 1.68e4T^{2} \)
11 \( 1 - 88.0T + 1.61e5T^{2} \)
13 \( 1 + (-307. + 533. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-285. - 494. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (1.21e3 - 2.10e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-1.14e3 + 1.97e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 3.68e3T + 2.86e7T^{2} \)
37 \( 1 - 3.06e3T + 6.93e7T^{2} \)
41 \( 1 + (1.24e3 + 2.15e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (2.45e3 + 4.24e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-8.78e3 + 1.52e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.27e4 - 2.20e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.17e4 + 2.03e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-9.88e3 + 1.71e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.36e4 - 2.37e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (1.67e4 + 2.90e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (8.98e3 + 1.55e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-4.12e4 - 7.14e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 4.02e4T + 3.93e9T^{2} \)
89 \( 1 + (2.76e4 - 4.78e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-1.33e4 - 2.31e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.52678356440831143604125171667, −12.08257948549961271204195489277, −11.24286691427313888874515086174, −10.59951055418158142161793610693, −8.221944112560468073597771907765, −7.40895267404300825358644383672, −6.23325074093675617149715439486, −5.23982290576784461304003821002, −2.29702101452553905620790743045, −1.14267990871421927148263978968, 1.27183755711022237125035262163, 4.29526914954635896709851362426, 4.92578575697943799790073986705, 5.93170109620680101994480269958, 8.382520116424340025987621466618, 9.282467381053617807084390497282, 10.41169088902570924008888825624, 11.43165069052284065014999724326, 12.12263539733929388099485759718, 14.06021484697014015035121042920

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