Properties

Label 2-76-19.7-c7-0-3
Degree $2$
Conductor $76$
Sign $0.832 + 0.554i$
Analytic cond. $23.7412$
Root an. cond. $4.87250$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−34.7 − 60.1i)3-s + (55.3 + 95.9i)5-s + 314.·7-s + (−1.31e3 + 2.27e3i)9-s + 5.49e3·11-s + (−6.97e3 + 1.20e4i)13-s + (3.84e3 − 6.66e3i)15-s + (7.10e3 + 1.23e4i)17-s + (4.43e3 + 2.95e4i)19-s + (−1.09e4 − 1.88e4i)21-s + (5.50e4 − 9.52e4i)23-s + (3.29e4 − 5.70e4i)25-s + 3.09e4·27-s + (8.12e4 − 1.40e5i)29-s − 1.62e4·31-s + ⋯
L(s)  = 1  + (−0.742 − 1.28i)3-s + (0.198 + 0.343i)5-s + 0.346·7-s + (−0.601 + 1.04i)9-s + 1.24·11-s + (−0.880 + 1.52i)13-s + (0.294 − 0.509i)15-s + (0.350 + 0.607i)17-s + (0.148 + 0.988i)19-s + (−0.257 − 0.445i)21-s + (0.942 − 1.63i)23-s + (0.421 − 0.729i)25-s + 0.302·27-s + (0.618 − 1.07i)29-s − 0.0979·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(23.7412\)
Root analytic conductor: \(4.87250\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :7/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.57109 - 0.475286i\)
\(L(\frac12)\) \(\approx\) \(1.57109 - 0.475286i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-4.43e3 - 2.95e4i)T \)
good3 \( 1 + (34.7 + 60.1i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-55.3 - 95.9i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 - 314.T + 8.23e5T^{2} \)
11 \( 1 - 5.49e3T + 1.94e7T^{2} \)
13 \( 1 + (6.97e3 - 1.20e4i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-7.10e3 - 1.23e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
23 \( 1 + (-5.50e4 + 9.52e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-8.12e4 + 1.40e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + 1.62e4T + 2.75e10T^{2} \)
37 \( 1 - 3.09e5T + 9.49e10T^{2} \)
41 \( 1 + (-7.82e4 - 1.35e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (-2.60e5 - 4.50e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (-1.37e4 + 2.37e4i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (3.11e5 - 5.38e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-4.35e5 - 7.54e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.44e6 + 2.49e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.19e6 + 2.06e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-5.49e5 - 9.51e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + (2.33e5 + 4.03e5i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-3.50e6 - 6.06e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 1.93e6T + 2.71e13T^{2} \)
89 \( 1 + (3.27e6 - 5.66e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + (-9.65e5 - 1.67e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−12.66007662057760968651666297573, −12.00302871405255403126013572042, −11.11806449642176787545309767895, −9.613193216268349878769413572098, −8.101848446483000836794877889196, −6.76243760149713490457385835169, −6.26283957023794563254966422390, −4.44534276551543698073604456770, −2.16684064658485241684613150971, −0.981135889361308549780183988939, 0.863851999984749370434135308312, 3.30633643316194640521711616050, 4.84823341883251469714100246465, 5.48625465916613203763122614869, 7.26297373540407154898452001537, 9.046100551067609950708434874074, 9.802432157410888492614915052250, 10.96319506879720151817201109934, 11.77865906360317064495462498651, 13.06422161062906924999741526172

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