↑

Properties

Label 2-28e2-28.27-c5-0-41
Degree $2$
Conductor $784$
Sign $0.156 - 0.987i$
Analytic cond. $125.740$
Root an. cond. $11.2134$
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  + 24.0·3-s + 1.38i·5-s + 336.·9-s + 278. i·11-s − 96.6i·13-s + 33.3i·15-s + 1.91e3i·17-s + 1.42e3·19-s + 499. i·23-s + 3.12e3·25-s + 2.26e3·27-s − 5.15e3·29-s − 4.52e3·31-s + 6.70e3i·33-s − 1.88e3·37-s + ⋯
L(s)  = 1  + 1.54·3-s + 0.0247i·5-s + 1.38·9-s + 0.694i·11-s − 0.158i·13-s + 0.0383i·15-s + 1.60i·17-s + 0.903·19-s + 0.196i·23-s + 0.999·25-s + 0.596·27-s − 1.13·29-s − 0.845·31-s + 1.07i·33-s − 0.226·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(125.740\)
Root analytic conductor: \(11.2134\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :5/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.671930213\)
\(L(\frac12)\) \(\approx\) \(3.671930213\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 24.0T + 243T^{2} \)
5 \( 1 - 1.38iT - 3.12e3T^{2} \)
11 \( 1 - 278. iT - 1.61e5T^{2} \)
13 \( 1 + 96.6iT - 3.71e5T^{2} \)
17 \( 1 - 1.91e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.42e3T + 2.47e6T^{2} \)
23 \( 1 - 499. iT - 6.43e6T^{2} \)
29 \( 1 + 5.15e3T + 2.05e7T^{2} \)
31 \( 1 + 4.52e3T + 2.86e7T^{2} \)
37 \( 1 + 1.88e3T + 6.93e7T^{2} \)
41 \( 1 - 2.13e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.43e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.41e4T + 2.29e8T^{2} \)
53 \( 1 + 7.35e3T + 4.18e8T^{2} \)
59 \( 1 + 3.74e4T + 7.14e8T^{2} \)
61 \( 1 + 4.40e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.17e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.39e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.17e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.57e4iT - 3.07e9T^{2} \)
83 \( 1 - 8.25e4T + 3.93e9T^{2} \)
89 \( 1 - 7.62e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.09e5iT - 8.58e9T^{2} \)
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

−9.472670011762578358970745183626, −8.961345574338823334832236882640, −7.962289822181192180042461915429, −7.51852868957560538236546981600, −6.41152801339914291858835822870, −5.16303480683497829446248021606, −3.97308425410903151529243882267, −3.29088636262453406976321873363, −2.21065457886317025221333925730, −1.35230267685653059034195785507, 0.55929600017544514009868884286, 1.87113820530599078710676316457, 2.92816217400479290291841464872, 3.49304323856557400981256771964, 4.70466607460110744249209437326, 5.77275266037414646347137969186, 7.25068808656752231330000006091, 7.51416885376209236517751313675, 8.863407050625200607781015980320, 8.975461540257059755074889849045

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