Properties

Label 2-76-19.10-c6-0-2
Degree $2$
Conductor $76$
Sign $0.999 + 0.0330i$
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  + (−31.3 − 37.3i)3-s + (37.0 + 13.4i)5-s + (−148. + 257. i)7-s + (−285. + 1.62e3i)9-s + (−242. − 419. i)11-s + (−423. + 504. i)13-s + (−657. − 1.80e3i)15-s + (332. + 1.88e3i)17-s + (6.81e3 − 809. i)19-s + (1.42e4 − 2.51e3i)21-s + (1.32e4 − 4.83e3i)23-s + (−1.07e4 − 9.04e3i)25-s + (3.87e4 − 2.23e4i)27-s + (1.88e4 + 3.32e3i)29-s + (2.07e3 + 1.20e3i)31-s + ⋯
L(s)  = 1  + (−1.16 − 1.38i)3-s + (0.296 + 0.107i)5-s + (−0.432 + 0.749i)7-s + (−0.392 + 2.22i)9-s + (−0.181 − 0.315i)11-s + (−0.192 + 0.229i)13-s + (−0.194 − 0.535i)15-s + (0.0676 + 0.383i)17-s + (0.993 − 0.118i)19-s + (1.53 − 0.271i)21-s + (1.09 − 0.397i)23-s + (−0.689 − 0.578i)25-s + (1.96 − 1.13i)27-s + (0.772 + 0.136i)29-s + (0.0697 + 0.0402i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.999 + 0.0330i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ 0.999 + 0.0330i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.01500 - 0.0167766i\)
\(L(\frac12)\) \(\approx\) \(1.01500 - 0.0167766i\)
\(L(4)\) 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 \( 1 + (-6.81e3 + 809. i)T \)
good3 \( 1 + (31.3 + 37.3i)T + (-126. + 717. i)T^{2} \)
5 \( 1 + (-37.0 - 13.4i)T + (1.19e4 + 1.00e4i)T^{2} \)
7 \( 1 + (148. - 257. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (242. + 419. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (423. - 504. i)T + (-8.38e5 - 4.75e6i)T^{2} \)
17 \( 1 + (-332. - 1.88e3i)T + (-2.26e7 + 8.25e6i)T^{2} \)
23 \( 1 + (-1.32e4 + 4.83e3i)T + (1.13e8 - 9.51e7i)T^{2} \)
29 \( 1 + (-1.88e4 - 3.32e3i)T + (5.58e8 + 2.03e8i)T^{2} \)
31 \( 1 + (-2.07e3 - 1.20e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 7.81e4iT - 2.56e9T^{2} \)
41 \( 1 + (-7.74e3 - 9.22e3i)T + (-8.24e8 + 4.67e9i)T^{2} \)
43 \( 1 + (1.73e4 + 6.32e3i)T + (4.84e9 + 4.06e9i)T^{2} \)
47 \( 1 + (-3.18e4 + 1.80e5i)T + (-1.01e10 - 3.68e9i)T^{2} \)
53 \( 1 + (-9.10e4 - 2.50e5i)T + (-1.69e10 + 1.42e10i)T^{2} \)
59 \( 1 + (-1.84e4 + 3.24e3i)T + (3.96e10 - 1.44e10i)T^{2} \)
61 \( 1 + (-2.10e5 + 7.65e4i)T + (3.94e10 - 3.31e10i)T^{2} \)
67 \( 1 + (-3.20e5 - 5.64e4i)T + (8.50e10 + 3.09e10i)T^{2} \)
71 \( 1 + (9.18e4 - 2.52e5i)T + (-9.81e10 - 8.23e10i)T^{2} \)
73 \( 1 + (5.32e5 - 4.46e5i)T + (2.62e10 - 1.49e11i)T^{2} \)
79 \( 1 + (-3.29e5 - 3.92e5i)T + (-4.22e10 + 2.39e11i)T^{2} \)
83 \( 1 + (3.37e5 - 5.83e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-2.54e5 + 3.03e5i)T + (-8.62e10 - 4.89e11i)T^{2} \)
97 \( 1 + (-9.25e5 + 1.63e5i)T + (7.82e11 - 2.84e11i)T^{2} \)
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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

−13.08715564936320932625387242579, −12.17303618755532944818347445278, −11.45367661730620082768949556987, −10.14581578162859928485916536062, −8.493555847237890353426319394683, −7.11710557577368999789164772143, −6.18109615321113813084936558772, −5.22590479290156379069478360559, −2.54433742827768727203523462321, −0.987664243536103201646225387184, 0.58956712942355738797499005898, 3.48109976681127151299424593416, 4.78905776913589623082165791172, 5.77119965141389857974819134251, 7.21386656276129263277318212367, 9.321631589110244690333963491421, 10.01004739644021694356037380255, 10.92870977176317270492073989598, 11.92254995313115765682855435503, 13.22272330411588538497019855870

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