Properties

Label 2-76-76.75-c5-0-0
Degree $2$
Conductor $76$
Sign $-0.434 + 0.900i$
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  + (0.797 + 5.60i)2-s + 16.3·3-s + (−30.7 + 8.93i)4-s − 79.2·5-s + (13.0 + 91.7i)6-s + 58.5i·7-s + (−74.5 − 164. i)8-s + 25.6·9-s + (−63.2 − 443. i)10-s − 540. i·11-s + (−503. + 146. i)12-s + 72.5i·13-s + (−327. + 46.6i)14-s − 1.29e3·15-s + (864. − 548. i)16-s − 1.33e3·17-s + ⋯
L(s)  = 1  + (0.140 + 0.990i)2-s + 1.05·3-s + (−0.960 + 0.279i)4-s − 1.41·5-s + (0.148 + 1.04i)6-s + 0.451i·7-s + (−0.411 − 0.911i)8-s + 0.105·9-s + (−0.199 − 1.40i)10-s − 1.34i·11-s + (−1.00 + 0.293i)12-s + 0.119i·13-s + (−0.446 + 0.0636i)14-s − 1.49·15-s + (0.844 − 0.536i)16-s − 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ -0.434 + 0.900i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0562328 - 0.0896001i\)
\(L(\frac12)\) \(\approx\) \(0.0562328 - 0.0896001i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.797 - 5.60i)T \)
19 \( 1 + (1.05e3 - 1.16e3i)T \)
good3 \( 1 - 16.3T + 243T^{2} \)
5 \( 1 + 79.2T + 3.12e3T^{2} \)
7 \( 1 - 58.5iT - 1.68e4T^{2} \)
11 \( 1 + 540. iT - 1.61e5T^{2} \)
13 \( 1 - 72.5iT - 3.71e5T^{2} \)
17 \( 1 + 1.33e3T + 1.41e6T^{2} \)
23 \( 1 + 620. iT - 6.43e6T^{2} \)
29 \( 1 - 604. iT - 2.05e7T^{2} \)
31 \( 1 + 2.53e3T + 2.86e7T^{2} \)
37 \( 1 - 4.48e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.03e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.05e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.18e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.01e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.86e4T + 7.14e8T^{2} \)
61 \( 1 + 3.76e4T + 8.44e8T^{2} \)
67 \( 1 + 4.22e4T + 1.35e9T^{2} \)
71 \( 1 - 8.15e4T + 1.80e9T^{2} \)
73 \( 1 + 5.10e3T + 2.07e9T^{2} \)
79 \( 1 + 4.47e4T + 3.07e9T^{2} \)
83 \( 1 + 1.12e5iT - 3.93e9T^{2} \)
89 \( 1 - 1.11e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.78e5iT - 8.58e9T^{2} \)
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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

−14.45659993937907477853594601578, −13.54386841654981628819428355089, −12.33396340104990999455398732795, −11.05264696943345214453828909087, −9.018116035566872079360880569177, −8.446961735194228280774773139401, −7.61560168869448969974350462919, −6.08850448908247575394779901492, −4.26311178746963815183026953144, −3.16822475105627199844437166373, 0.03727464090700783001291290886, 2.24566649459319252353988107744, 3.69021717167392742982111954519, 4.57860196781859299525791487890, 7.25761493804113373755987765383, 8.363573810427507690446298511074, 9.321677012996508499407986747001, 10.70195091273842811251953357849, 11.69605801495828896110244803771, 12.76009059386672326989749393507

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