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Properties

Label 2-588-21.17-c3-0-29
Degree $2$
Conductor $588$
Sign $-0.00875 + 0.999i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.84 − 4.85i)3-s + (1.01 + 1.75i)5-s + (−20.1 − 17.9i)9-s + (25.2 + 14.5i)11-s + 25.7i·13-s + (10.4 − 1.67i)15-s + (−1.39 + 2.42i)17-s + (92.8 − 53.6i)19-s + (70.3 − 40.6i)23-s + (60.4 − 104. i)25-s + (−124. + 64.6i)27-s − 170. i·29-s + (22.4 + 12.9i)31-s + (117. − 95.7i)33-s + (−90.6 − 156. i)37-s + ⋯
L(s)  = 1  + (0.355 − 0.934i)3-s + (0.0907 + 0.157i)5-s + (−0.746 − 0.665i)9-s + (0.693 + 0.400i)11-s + 0.548i·13-s + (0.179 − 0.0288i)15-s + (−0.0199 + 0.0345i)17-s + (1.12 − 0.647i)19-s + (0.637 − 0.368i)23-s + (0.483 − 0.837i)25-s + (−0.887 + 0.460i)27-s − 1.09i·29-s + (0.130 + 0.0750i)31-s + (0.620 − 0.505i)33-s + (−0.402 − 0.697i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00875 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00875 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.00875 + 0.999i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.00875 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.227057978\)
\(L(\frac12)\) \(\approx\) \(2.227057978\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.84 + 4.85i)T \)
7 \( 1 \)
good5 \( 1 + (-1.01 - 1.75i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-25.2 - 14.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 25.7iT - 2.19e3T^{2} \)
17 \( 1 + (1.39 - 2.42i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-92.8 + 53.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-70.3 + 40.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 170. iT - 2.43e4T^{2} \)
31 \( 1 + (-22.4 - 12.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (90.6 + 156. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 257.T + 6.89e4T^{2} \)
43 \( 1 - 239.T + 7.95e4T^{2} \)
47 \( 1 + (186. + 323. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-489. - 282. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-189. + 328. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (194. - 112. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (167. - 290. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.02e3iT - 3.57e5T^{2} \)
73 \( 1 + (-505. - 292. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-497. - 861. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (738. + 1.27e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.22e3iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.903174723116107799558302485989, −9.089286423784348579234160814810, −8.325255382272602399080562991972, −7.16937144598307485733455517571, −6.75116746481267919556535007646, −5.64255691794579392330647108207, −4.32588791798401565609860763441, −3.04862643002557592750714504980, −1.96600795216194380406153453295, −0.69782598317606412278591545198, 1.25980433591629567282240354064, 2.99559338340090110107705544189, 3.70945558021346375253685069908, 4.99431502569886080038892520227, 5.65591073397402972253288334774, 6.98603537689426273535769456526, 8.063596871216955295782871550197, 8.916413681065644019344871961008, 9.567194269650770592247941739608, 10.43133368796373603662554724284

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