Properties

Label 2-588-7.2-c3-0-6
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 + (−8.32 + 14.4i)5-s + (−4.5 + 7.79i)9-s + (35.8 + 62.1i)11-s + 65.3·13-s − 49.9·15-s + (45.2 + 78.3i)17-s + (81.9 − 141. i)19-s + (−39.6 + 68.6i)23-s + (−76.1 − 131. i)25-s − 27·27-s − 43.2·29-s + (67.8 + 117. i)31-s + (−107. + 186. i)33-s + (−135. + 234. i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.744 + 1.29i)5-s + (−0.166 + 0.288i)9-s + (0.983 + 1.70i)11-s + 1.39·13-s − 0.860·15-s + (0.645 + 1.11i)17-s + (0.989 − 1.71i)19-s + (−0.359 + 0.622i)23-s + (−0.609 − 1.05i)25-s − 0.192·27-s − 0.277·29-s + (0.392 + 0.680i)31-s + (−0.567 + 0.983i)33-s + (−0.601 + 1.04i)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

\(L(2)\) \(\approx\) \(2.012391874\)
\(L(\frac12)\) \(\approx\) \(2.012391874\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.32 - 14.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-35.8 - 62.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 65.3T + 2.19e3T^{2} \)
17 \( 1 + (-45.2 - 78.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-81.9 + 141. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (39.6 - 68.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 43.2T + 2.43e4T^{2} \)
31 \( 1 + (-67.8 - 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (135. - 234. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 152.T + 6.89e4T^{2} \)
43 \( 1 + 177.T + 7.95e4T^{2} \)
47 \( 1 + (22.8 - 39.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-79.2 - 137. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (195. + 339. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-275. + 477. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (229. + 397. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 486.T + 3.57e5T^{2} \)
73 \( 1 + (-287. - 497. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-334. + 578. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 76.2T + 5.71e5T^{2} \)
89 \( 1 + (-683. + 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 242.T + 9.12e5T^{2} \)
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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

−10.60166591473853821868782639972, −9.877346428690698474160880169869, −8.972205199448086426584543086138, −7.916059187802519787871772689237, −7.04260762582776187951393405408, −6.38715399487303836498963555407, −4.87167118336730174398552187312, −3.78870729273058751963345047142, −3.18523220645407921408896935294, −1.62000174636289551726750867163, 0.66129641761407339704116190940, 1.32610421421308752356688826217, 3.37645157184291847832757874831, 3.96017238661141096889488410293, 5.47066629385577582687656750646, 6.14529610731108760029894113888, 7.49477236493606801895502280194, 8.425831463781855190704090695147, 8.663527509313537223970216901749, 9.702724401519868966593075232467

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