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Properties

Label 2-76-4.3-c4-0-30
Degree $2$
Conductor $76$
Sign $-0.694 + 0.719i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.49 − 3.70i)2-s − 6.26i·3-s + (−11.5 − 11.1i)4-s + 49.7·5-s + (−23.2 − 9.39i)6-s − 51.4i·7-s + (−58.4 + 26.0i)8-s + 41.6·9-s + (74.5 − 184. i)10-s + 67.5i·11-s + (−69.6 + 72.1i)12-s − 124.·13-s + (−190. − 77.1i)14-s − 311. i·15-s + (8.83 + 255. i)16-s − 274.·17-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s − 0.696i·3-s + (−0.719 − 0.694i)4-s + 1.99·5-s + (−0.645 − 0.261i)6-s − 1.05i·7-s + (−0.913 + 0.406i)8-s + 0.514·9-s + (0.745 − 1.84i)10-s + 0.557i·11-s + (−0.484 + 0.501i)12-s − 0.739·13-s + (−0.974 − 0.393i)14-s − 1.38i·15-s + (0.0345 + 0.999i)16-s − 0.950·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.694 + 0.719i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.943158 - 2.22253i\)
\(L(\frac12)\) \(\approx\) \(0.943158 - 2.22253i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.49 + 3.70i)T \)
19 \( 1 + 82.8iT \)
good3 \( 1 + 6.26iT - 81T^{2} \)
5 \( 1 - 49.7T + 625T^{2} \)
7 \( 1 + 51.4iT - 2.40e3T^{2} \)
11 \( 1 - 67.5iT - 1.46e4T^{2} \)
13 \( 1 + 124.T + 2.85e4T^{2} \)
17 \( 1 + 274.T + 8.35e4T^{2} \)
23 \( 1 - 914. iT - 2.79e5T^{2} \)
29 \( 1 + 414.T + 7.07e5T^{2} \)
31 \( 1 + 823. iT - 9.23e5T^{2} \)
37 \( 1 - 1.91e3T + 1.87e6T^{2} \)
41 \( 1 - 457.T + 2.82e6T^{2} \)
43 \( 1 - 1.52e3iT - 3.41e6T^{2} \)
47 \( 1 - 742. iT - 4.87e6T^{2} \)
53 \( 1 - 1.33e3T + 7.89e6T^{2} \)
59 \( 1 - 2.25e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.11e3T + 1.38e7T^{2} \)
67 \( 1 + 7.02e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.13e3iT - 2.54e7T^{2} \)
73 \( 1 + 578.T + 2.83e7T^{2} \)
79 \( 1 - 373. iT - 3.89e7T^{2} \)
83 \( 1 + 548. iT - 4.74e7T^{2} \)
89 \( 1 + 4.82e3T + 6.27e7T^{2} \)
97 \( 1 + 4.83e3T + 8.85e7T^{2} \)
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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

−13.36486833062611606447056292199, −12.67688933168330164655928219373, −11.08325533177863098306021391341, −9.913687314444618812518780114559, −9.464052659381407195081733474038, −7.23770963868883609463784086128, −5.98723946361149330256650032436, −4.55554894401694927398995657562, −2.32847098502824241523128422803, −1.28430133050614145339040122929, 2.51563838756778938675209629784, 4.73510480352172181945050738573, 5.71860082715531259611251060804, 6.68503148473613897527638712269, 8.760304483159997136480557259971, 9.414751814351762654353138909837, 10.48211009155093148662366451081, 12.47282733325323660200949024258, 13.27907420254542659630149420930, 14.36288630240301826961833160277

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