Properties

Label 2-76-19.12-c6-0-6
Degree $2$
Conductor $76$
Sign $0.00985 + 0.999i$
Analytic cond. $17.4841$
Root an. cond. $4.18140$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−20.4 + 11.8i)3-s + (15.3 + 26.6i)5-s − 77.2·7-s + (−85.9 + 148. i)9-s − 1.57e3·11-s + (2.81e3 + 1.62e3i)13-s + (−628. − 362. i)15-s + (−3.36e3 − 5.82e3i)17-s + (2.77e3 − 6.27e3i)19-s + (1.57e3 − 911. i)21-s + (2.29e3 − 3.96e3i)23-s + (7.34e3 − 1.27e4i)25-s − 2.12e4i·27-s + (−2.26e4 − 1.30e4i)29-s − 2.58e4i·31-s + ⋯
L(s)  = 1  + (−0.757 + 0.437i)3-s + (0.122 + 0.212i)5-s − 0.225·7-s + (−0.117 + 0.204i)9-s − 1.18·11-s + (1.28 + 0.740i)13-s + (−0.186 − 0.107i)15-s + (−0.684 − 1.18i)17-s + (0.403 − 0.914i)19-s + (0.170 − 0.0984i)21-s + (0.188 − 0.326i)23-s + (0.469 − 0.813i)25-s − 1.08i·27-s + (−0.927 − 0.535i)29-s − 0.866i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00985 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.00985 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.00985 + 0.999i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ 0.00985 + 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.431585 - 0.427353i\)
\(L(\frac12)\) \(\approx\) \(0.431585 - 0.427353i\)
\(L(4)\) 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 \( 1 + (-2.77e3 + 6.27e3i)T \)
good3 \( 1 + (20.4 - 11.8i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-15.3 - 26.6i)T + (-7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + 77.2T + 1.17e5T^{2} \)
11 \( 1 + 1.57e3T + 1.77e6T^{2} \)
13 \( 1 + (-2.81e3 - 1.62e3i)T + (2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (3.36e3 + 5.82e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
23 \( 1 + (-2.29e3 + 3.96e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.26e4 + 1.30e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + 2.58e4iT - 8.87e8T^{2} \)
37 \( 1 - 8.33e4iT - 2.56e9T^{2} \)
41 \( 1 + (-8.11e4 + 4.68e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (5.73e4 + 9.93e4i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-9.48e3 + 1.64e4i)T + (-5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-5.96e4 - 3.44e4i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-1.77e4 + 1.02e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-3.94e4 + 6.82e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-2.37e4 - 1.37e4i)T + (4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (3.45e5 - 1.99e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (-2.77e5 - 4.79e5i)T + (-7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (4.24e5 - 2.44e5i)T + (1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 5.16e5T + 3.26e11T^{2} \)
89 \( 1 + (7.10e5 + 4.10e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + (-4.29e5 + 2.47e5i)T + (4.16e11 - 7.21e11i)T^{2} \)
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

−13.18730885019602108892201189434, −11.54132700400200711511412603333, −10.97730858854676913424522597179, −9.837899983887860141184430915491, −8.500346579415412463336746423391, −6.93722864725360153593430980784, −5.67105793743717352037663720005, −4.50462532793418828019680227160, −2.59829953324573887032163298306, −0.26893030853151177977659685327, 1.31913763456951994581828109545, 3.42914099478688980213864863178, 5.41162611550192434534785186829, 6.21831416734815889278351711688, 7.72101831271850779795620562156, 8.958392612903654836386235062914, 10.53076468418124350125117441864, 11.26095791437808135439056682104, 12.86910103400671562933987726356, 12.96404653592905524853783513624

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