Properties

Label 2-76-76.75-c5-0-20
Degree $2$
Conductor $76$
Sign $0.981 - 0.189i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.877 − 5.58i)2-s + 10.0·3-s + (−30.4 + 9.80i)4-s + 80.9·5-s + (−8.82 − 56.2i)6-s + 200. i·7-s + (81.4 + 161. i)8-s − 141.·9-s + (−70.9 − 452. i)10-s + 464. i·11-s + (−306. + 98.6i)12-s + 659. i·13-s + (1.11e3 − 175. i)14-s + 814.·15-s + (831. − 597. i)16-s − 1.15e3·17-s + ⋯
L(s)  = 1  + (−0.155 − 0.987i)2-s + 0.645·3-s + (−0.951 + 0.306i)4-s + 1.44·5-s + (−0.100 − 0.638i)6-s + 1.54i·7-s + (0.450 + 0.892i)8-s − 0.582·9-s + (−0.224 − 1.42i)10-s + 1.15i·11-s + (−0.614 + 0.197i)12-s + 1.08i·13-s + (1.52 − 0.239i)14-s + 0.934·15-s + (0.812 − 0.583i)16-s − 0.971·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.981 - 0.189i$
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: \((2,\ 76,\ (\ :5/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.05913 + 0.197340i\)
\(L(\frac12)\) \(\approx\) \(2.05913 + 0.197340i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.877 + 5.58i)T \)
19 \( 1 + (-1.56e3 - 188. i)T \)
good3 \( 1 - 10.0T + 243T^{2} \)
5 \( 1 - 80.9T + 3.12e3T^{2} \)
7 \( 1 - 200. iT - 1.68e4T^{2} \)
11 \( 1 - 464. iT - 1.61e5T^{2} \)
13 \( 1 - 659. iT - 3.71e5T^{2} \)
17 \( 1 + 1.15e3T + 1.41e6T^{2} \)
23 \( 1 + 2.53e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.62e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.00e4T + 2.86e7T^{2} \)
37 \( 1 - 2.10e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.12e3iT - 1.15e8T^{2} \)
43 \( 1 - 575. iT - 1.47e8T^{2} \)
47 \( 1 - 1.34e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.83e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.85e4T + 7.14e8T^{2} \)
61 \( 1 - 2.50e4T + 8.44e8T^{2} \)
67 \( 1 + 3.21e3T + 1.35e9T^{2} \)
71 \( 1 - 1.08e4T + 1.80e9T^{2} \)
73 \( 1 - 8.06e4T + 2.07e9T^{2} \)
79 \( 1 - 2.18e4T + 3.07e9T^{2} \)
83 \( 1 - 3.29e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.23e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.55e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.58940903094489265016549377363, −12.39068968267676170063694182375, −11.50433368531333640659920488834, −9.808115982351566727828099843018, −9.326107796172541065889783182313, −8.398801363544848354760823158246, −6.17911080691765832914396429556, −4.78884301418363545718519053809, −2.54107972432135007682191123078, −2.05905274532599657150640688783, 0.915356703630710306724642235058, 3.30945151276424772782890196605, 5.24349800139769285844115027198, 6.36293157842522966466595970687, 7.69639653280329339548079564690, 8.819322747959241019071629320520, 9.868156258567910611528996489260, 10.83098769407002661246818018999, 13.31634462018297126874684993299, 13.69170524208848802405449094426

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