Properties

Label 2-288-4.3-c4-0-10
Degree $2$
Conductor $288$
Sign $0.707 + 0.707i$
Analytic cond. $29.7705$
Root an. cond. $5.45623$
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  − 34.8·5-s + 73.6i·7-s − 171. i·11-s − 288.·13-s + 197.·17-s + 83.3i·19-s + 515. i·23-s + 588.·25-s + 1.22e3·29-s − 426. i·31-s − 2.56e3i·35-s + 1.40e3·37-s + 1.01e3·41-s − 2.70e3i·43-s − 4.27e3i·47-s + ⋯
L(s)  = 1  − 1.39·5-s + 1.50i·7-s − 1.41i·11-s − 1.70·13-s + 0.683·17-s + 0.230i·19-s + 0.975i·23-s + 0.941·25-s + 1.45·29-s − 0.443i·31-s − 2.09i·35-s + 1.02·37-s + 0.603·41-s − 1.46i·43-s − 1.93i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(29.7705\)
Root analytic conductor: \(5.45623\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :2),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9390684538\)
\(L(\frac12)\) \(\approx\) \(0.9390684538\)
\(L(3)\) 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 \( 1 \)
good5 \( 1 + 34.8T + 625T^{2} \)
7 \( 1 - 73.6iT - 2.40e3T^{2} \)
11 \( 1 + 171. iT - 1.46e4T^{2} \)
13 \( 1 + 288.T + 2.85e4T^{2} \)
17 \( 1 - 197.T + 8.35e4T^{2} \)
19 \( 1 - 83.3iT - 1.30e5T^{2} \)
23 \( 1 - 515. iT - 2.79e5T^{2} \)
29 \( 1 - 1.22e3T + 7.07e5T^{2} \)
31 \( 1 + 426. iT - 9.23e5T^{2} \)
37 \( 1 - 1.40e3T + 1.87e6T^{2} \)
41 \( 1 - 1.01e3T + 2.82e6T^{2} \)
43 \( 1 + 2.70e3iT - 3.41e6T^{2} \)
47 \( 1 + 4.27e3iT - 4.87e6T^{2} \)
53 \( 1 - 854.T + 7.89e6T^{2} \)
59 \( 1 - 705. iT - 1.21e7T^{2} \)
61 \( 1 + 537.T + 1.38e7T^{2} \)
67 \( 1 - 2.04e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.20e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.17e3T + 2.83e7T^{2} \)
79 \( 1 + 124. iT - 3.89e7T^{2} \)
83 \( 1 + 1.86e3iT - 4.74e7T^{2} \)
89 \( 1 + 9.86e3T + 6.27e7T^{2} \)
97 \( 1 - 1.56e4T + 8.85e7T^{2} \)
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

−11.42420938906225502951742433256, −10.09011093752731616137692316834, −8.950946303370324401027777590515, −8.181647756939389672757416333421, −7.36537120629971482292093160155, −5.92143057774019888002533238730, −5.00345718495702324384080358894, −3.57171653056850152002051487610, −2.54489176838542221683344611287, −0.41455717528071116592621064081, 0.850732300979571746431442935904, 2.82933724005438392660666299738, 4.38882828583970856082423825534, 4.58880018203434786875419547981, 6.78260945635726946282796911538, 7.47578972644485694164993028493, 7.991167914924184180547721213778, 9.652504836337229148201519777305, 10.29353430012897390066703804951, 11.30974513212272093156615005456

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