Properties

Label 2-76-19.3-c4-0-6
Degree $2$
Conductor $76$
Sign $-0.999 + 0.0208i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.65 − 12.7i)3-s + (6.59 − 37.3i)5-s + (5.00 + 8.66i)7-s + (−79.9 + 67.0i)9-s + (−7.63 + 13.2i)11-s + (−22.1 + 60.9i)13-s + (−508. + 89.7i)15-s + (−348. − 292. i)17-s + (281. + 226. i)19-s + (87.5 − 104. i)21-s + (49.2 + 279. i)23-s + (−766. − 278. i)25-s + (275. + 158. i)27-s + (−472. − 562. i)29-s + (1.43e3 − 825. i)31-s + ⋯
L(s)  = 1  + (−0.517 − 1.42i)3-s + (0.263 − 1.49i)5-s + (0.102 + 0.176i)7-s + (−0.986 + 0.828i)9-s + (−0.0631 + 0.109i)11-s + (−0.131 + 0.360i)13-s + (−2.26 + 0.398i)15-s + (−1.20 − 1.01i)17-s + (0.778 + 0.627i)19-s + (0.198 − 0.236i)21-s + (0.0931 + 0.528i)23-s + (−1.22 − 0.446i)25-s + (0.377 + 0.218i)27-s + (−0.561 − 0.669i)29-s + (1.48 − 0.859i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0208i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.999 + 0.0208i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.999 + 0.0208i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0113965 - 1.09441i\)
\(L(\frac12)\) \(\approx\) \(0.0113965 - 1.09441i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-281. - 226. i)T \)
good3 \( 1 + (4.65 + 12.7i)T + (-62.0 + 52.0i)T^{2} \)
5 \( 1 + (-6.59 + 37.3i)T + (-587. - 213. i)T^{2} \)
7 \( 1 + (-5.00 - 8.66i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (7.63 - 13.2i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (22.1 - 60.9i)T + (-2.18e4 - 1.83e4i)T^{2} \)
17 \( 1 + (348. + 292. i)T + (1.45e4 + 8.22e4i)T^{2} \)
23 \( 1 + (-49.2 - 279. i)T + (-2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (472. + 562. i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (-1.43e3 + 825. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.47e3iT - 1.87e6T^{2} \)
41 \( 1 + (842. + 2.31e3i)T + (-2.16e6 + 1.81e6i)T^{2} \)
43 \( 1 + (329. - 1.86e3i)T + (-3.21e6 - 1.16e6i)T^{2} \)
47 \( 1 + (-985. + 827. i)T + (8.47e5 - 4.80e6i)T^{2} \)
53 \( 1 + (-3.11e3 + 549. i)T + (7.41e6 - 2.69e6i)T^{2} \)
59 \( 1 + (-938. + 1.11e3i)T + (-2.10e6 - 1.19e7i)T^{2} \)
61 \( 1 + (920. + 5.21e3i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (4.06e3 + 4.84e3i)T + (-3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (9.08e3 + 1.60e3i)T + (2.38e7 + 8.69e6i)T^{2} \)
73 \( 1 + (3.63e3 - 1.32e3i)T + (2.17e7 - 1.82e7i)T^{2} \)
79 \( 1 + (1.43e3 + 3.95e3i)T + (-2.98e7 + 2.50e7i)T^{2} \)
83 \( 1 + (-6.21e3 - 1.07e4i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (-4.13e3 + 1.13e4i)T + (-4.80e7 - 4.03e7i)T^{2} \)
97 \( 1 + (1.90e3 - 2.27e3i)T + (-1.53e7 - 8.71e7i)T^{2} \)
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

−13.32208094269705319411850722975, −12.07732760633665001709209166086, −11.65192214063342309728413070978, −9.620710013114885651902094530992, −8.468001887393485787815262481608, −7.30777010116439045169279678209, −5.97389159811509695414267302770, −4.79498963206766465375998566934, −1.92754594308693210972277858931, −0.59713940044553799632020441467, 2.93576973296331298412622297448, 4.34854039272836073712435032062, 5.83196871560183604890904480523, 7.07263654082708889470181704482, 8.936949508719686204837811609169, 10.36834068246534644821834464617, 10.59437478786967578598176000742, 11.64754354423140989756709006715, 13.44291260680767230032841053897, 14.67093745061315908503663205389

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