↑

Properties

Label 2-684-19.12-c4-0-17
Degree $2$
Conductor $684$
Sign $0.561 - 0.827i$
Analytic cond. $70.7050$
Root an. cond. $8.40862$
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  + (18.6 + 32.2i)5-s − 25.9·7-s + 189.·11-s + (111. + 64.4i)13-s + (160. + 277. i)17-s + (−8.91 − 360. i)19-s + (294. − 510. i)23-s + (−379. + 657. i)25-s + (788. + 455. i)29-s − 1.65e3i·31-s + (−481. − 834. i)35-s + 1.02e3i·37-s + (1.38e3 − 802. i)41-s + (905. + 1.56e3i)43-s + (516. − 894. i)47-s + ⋯
L(s)  = 1  + (0.744 + 1.28i)5-s − 0.528·7-s + 1.56·11-s + (0.660 + 0.381i)13-s + (0.554 + 0.960i)17-s + (−0.0247 − 0.999i)19-s + (0.556 − 0.964i)23-s + (−0.607 + 1.05i)25-s + (0.938 + 0.541i)29-s − 1.72i·31-s + (−0.393 − 0.681i)35-s + 0.748i·37-s + (0.826 − 0.477i)41-s + (0.489 + 0.847i)43-s + (0.233 − 0.404i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(70.7050\)
Root analytic conductor: \(8.40862\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :2),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.904583809\)
\(L(\frac12)\) \(\approx\) \(2.904583809\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (8.91 + 360. i)T \)
good5 \( 1 + (-18.6 - 32.2i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 25.9T + 2.40e3T^{2} \)
11 \( 1 - 189.T + 1.46e4T^{2} \)
13 \( 1 + (-111. - 64.4i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-160. - 277. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-294. + 510. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-788. - 455. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.65e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.02e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.38e3 + 802. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-905. - 1.56e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-516. + 894. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-656. - 379. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (896. - 517. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.76e3 + 4.79e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.01e3 - 585. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (6.05e3 - 3.49e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-3.09e3 - 5.36e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-4.44e3 + 2.56e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 3.97e3T + 4.74e7T^{2} \)
89 \( 1 + (7.93e3 + 4.58e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (9.60e3 - 5.54e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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

−9.979980002600335253373650290524, −9.334636997352210304846050762286, −8.427863320535960902205019623485, −7.02944525002132849962528038805, −6.47433910442011356241773562035, −5.95859214047417846365539976886, −4.32821337381312263436195079410, −3.35824430047617986048326657073, −2.36816080850067682826297733096, −1.07252855079135378333617207867, 0.880722120035386599770524706555, 1.51576430285354677045395199867, 3.18644911000787719707027850306, 4.22926623917868590168959898311, 5.35011899410107159004624483752, 6.03543679227568197751674425965, 7.01919036451333639408393284479, 8.244613783566041892854304066272, 9.125783718318665676664404763620, 9.480043535209448111122299983555

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