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Properties

Label 2-76-76.75-c5-0-27
Degree $2$
Conductor $76$
Sign $0.0685 - 0.997i$
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  + (2.99 + 4.79i)2-s + 24.6·3-s + (−14.0 + 28.7i)4-s + 37.3·5-s + (73.8 + 118. i)6-s + 35.5i·7-s + (−180. + 19.1i)8-s + 362.·9-s + (112. + 179. i)10-s + 184. i·11-s + (−344. + 708. i)12-s − 119. i·13-s + (−170. + 106. i)14-s + 919.·15-s + (−631. − 806. i)16-s + 152.·17-s + ⋯
L(s)  = 1  + (0.530 + 0.847i)2-s + 1.57·3-s + (−0.437 + 0.899i)4-s + 0.668·5-s + (0.837 + 1.33i)6-s + 0.274i·7-s + (−0.994 + 0.105i)8-s + 1.49·9-s + (0.354 + 0.566i)10-s + 0.460i·11-s + (−0.691 + 1.41i)12-s − 0.195i·13-s + (−0.232 + 0.145i)14-s + 1.05·15-s + (−0.616 − 0.787i)16-s + 0.128·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0685 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0685 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.0685 - 0.997i$
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: \((2,\ 76,\ (\ :5/2),\ 0.0685 - 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.86006 + 2.67034i\)
\(L(\frac12)\) \(\approx\) \(2.86006 + 2.67034i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.99 - 4.79i)T \)
19 \( 1 + (-1.45e3 + 590. i)T \)
good3 \( 1 - 24.6T + 243T^{2} \)
5 \( 1 - 37.3T + 3.12e3T^{2} \)
7 \( 1 - 35.5iT - 1.68e4T^{2} \)
11 \( 1 - 184. iT - 1.61e5T^{2} \)
13 \( 1 + 119. iT - 3.71e5T^{2} \)
17 \( 1 - 152.T + 1.41e6T^{2} \)
23 \( 1 + 1.58e3iT - 6.43e6T^{2} \)
29 \( 1 + 54.3iT - 2.05e7T^{2} \)
31 \( 1 + 4.26e3T + 2.86e7T^{2} \)
37 \( 1 - 1.21e4iT - 6.93e7T^{2} \)
41 \( 1 - 5.66e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.92e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.66e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.94e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.28e4T + 7.14e8T^{2} \)
61 \( 1 + 3.17e4T + 8.44e8T^{2} \)
67 \( 1 + 2.17e4T + 1.35e9T^{2} \)
71 \( 1 - 1.28e4T + 1.80e9T^{2} \)
73 \( 1 - 3.33e4T + 2.07e9T^{2} \)
79 \( 1 - 5.70e4T + 3.07e9T^{2} \)
83 \( 1 + 1.47e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.38e5iT - 5.58e9T^{2} \)
97 \( 1 + 3.42e4iT - 8.58e9T^{2} \)
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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

−13.80133862849721948329845720661, −13.32873637536661224429031113122, −12.08190068554500906002231670602, −9.898859961641845001481134361970, −8.989872554269670059830524757745, −8.014707343704622621485065529292, −6.88054720323801012148626080310, −5.26339516599615073523205377574, −3.63419588321013005469997104747, −2.32567614021961657357644346551, 1.52833004744630371630509578575, 2.83619146338179397092089613441, 3.96876391425675359935074226359, 5.75326664365509834178344437933, 7.64281014500979642719763492745, 9.115950629366677791296087245544, 9.691700804308190016730305773439, 10.97959951426109983911775265650, 12.47660494547186866866412304763, 13.61779389073935612297292641838

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