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Properties

Label 2-588-7.2-c3-0-7
Degree $2$
Conductor $588$
Sign $0.827 + 0.561i$
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  + (−1.5 − 2.59i)3-s + (−9.57 + 16.5i)5-s + (−4.5 + 7.79i)9-s + (−20.2 − 35.1i)11-s − 50.4·13-s + 57.4·15-s + (−25.9 − 44.9i)17-s + (−16.5 + 28.6i)19-s + (−31.4 + 54.4i)23-s + (−120. − 209. i)25-s + 27·27-s + 129.·29-s + (121. + 209. i)31-s + (−60.8 + 105. i)33-s + (194. − 337. i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.855 + 1.48i)5-s + (−0.166 + 0.288i)9-s + (−0.556 − 0.963i)11-s − 1.07·13-s + 0.988·15-s + (−0.370 − 0.641i)17-s + (−0.200 + 0.346i)19-s + (−0.284 + 0.493i)23-s + (−0.965 − 1.67i)25-s + 0.192·27-s + 0.831·29-s + (0.702 + 1.21i)31-s + (−0.321 + 0.556i)33-s + (0.865 − 1.49i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8943646064\)
\(L(\frac12)\) \(\approx\) \(0.8943646064\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (9.57 - 16.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (20.2 + 35.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 50.4T + 2.19e3T^{2} \)
17 \( 1 + (25.9 + 44.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (16.5 - 28.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (31.4 - 54.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 129.T + 2.43e4T^{2} \)
31 \( 1 + (-121. - 209. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-194. + 337. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 470.T + 6.89e4T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + (193. - 334. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-305. - 529. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-113. - 196. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-362. + 628. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (522. + 905. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 169.T + 3.57e5T^{2} \)
73 \( 1 + (190. + 330. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-580. + 1.00e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 808.T + 5.71e5T^{2} \)
89 \( 1 + (159. - 277. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.13e3T + 9.12e5T^{2} \)
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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

−10.60708094783437873541443746272, −9.406774848436484258202103329403, −8.049058540499923076294412484168, −7.53323877740115200939858037110, −6.71106079482243209450930936521, −5.84735878888042806694305874460, −4.53573275004226536856714343039, −3.19719909154563662211379898187, −2.46881131789448269087727383969, −0.42726525883454178519683969220, 0.71903449962457479139086214728, 2.41971479902186203164371785047, 4.20836683470708206835568496147, 4.58118643617998428429222786486, 5.46545408572012833853052225808, 6.83807013343624193301219482042, 7.965199573812991088149011493505, 8.510640845780545822366469572048, 9.623657658102995425580038986989, 10.15838362871816207259883492939

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