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Properties

Label 2-76-19.7-c5-0-7
Degree $2$
Conductor $76$
Sign $-0.952 - 0.302i$
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  + (−6.79 − 11.7i)3-s + (−47.4 − 82.2i)5-s + 189.·7-s + (29.1 − 50.4i)9-s − 530.·11-s + (−353. + 612. i)13-s + (−645. + 1.11e3i)15-s + (−764. − 1.32e3i)17-s + (654. + 1.43e3i)19-s + (−1.29e3 − 2.23e3i)21-s + (497. − 861. i)23-s + (−2.94e3 + 5.10e3i)25-s − 4.09e3·27-s + (−1.29e3 + 2.23e3i)29-s − 2.79e3·31-s + ⋯
L(s)  = 1  + (−0.435 − 0.755i)3-s + (−0.849 − 1.47i)5-s + 1.46·7-s + (0.119 − 0.207i)9-s − 1.32·11-s + (−0.580 + 1.00i)13-s + (−0.740 + 1.28i)15-s + (−0.641 − 1.11i)17-s + (0.415 + 0.909i)19-s + (−0.638 − 1.10i)21-s + (0.196 − 0.339i)23-s + (−0.943 + 1.63i)25-s − 1.08·27-s + (−0.284 + 0.493i)29-s − 0.521·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.302i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.952 - 0.302i$
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: \((2,\ 76,\ (\ :5/2),\ -0.952 - 0.302i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.108914 + 0.702079i\)
\(L(\frac12)\) \(\approx\) \(0.108914 + 0.702079i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-654. - 1.43e3i)T \)
good3 \( 1 + (6.79 + 11.7i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (47.4 + 82.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 189.T + 1.68e4T^{2} \)
11 \( 1 + 530.T + 1.61e5T^{2} \)
13 \( 1 + (353. - 612. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (764. + 1.32e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (-497. + 861. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.29e3 - 2.23e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 2.79e3T + 2.86e7T^{2} \)
37 \( 1 - 7.23e3T + 6.93e7T^{2} \)
41 \( 1 + (-2.18e3 - 3.77e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (3.12e3 + 5.40e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.23e4 + 2.13e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.54e4 - 2.68e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.81e4 + 3.14e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.88e3 + 5.00e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.03e4 + 5.26e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.48e4 - 2.56e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (3.90e4 + 6.75e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.34e4 + 5.78e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 2.19e4T + 3.93e9T^{2} \)
89 \( 1 + (4.53e4 - 7.85e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (3.07e4 + 5.32e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.65492764127384811536268804876, −11.96495070475649483396462622189, −11.16482623699728172029757701181, −9.256290197808679993374197264663, −8.065885714793775470722504191405, −7.32881418130175659718633947682, −5.29226609845188516662585121774, −4.47437689967598862417035965449, −1.69920036523747989564474223248, −0.32333072983143451933045183540, 2.60811032359284983998354109445, 4.30667355643831237281849406181, 5.46596794794189857591978576356, 7.41427218722637340255457054945, 8.044015669145436853515927713018, 10.15858404293289324053675105877, 10.98091189181781995708883968755, 11.29997678165821662814364385007, 13.04002051027462330187024970979, 14.51798548860852621013957904346

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