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Properties

Label 2-588-4.3-c2-0-10
Degree $2$
Conductor $588$
Sign $-0.999 + 0.0161i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.42 + 1.40i)2-s − 1.73i·3-s + (0.0647 + 3.99i)4-s − 1.94·5-s + (2.42 − 2.46i)6-s + (−5.51 + 5.79i)8-s − 2.99·9-s + (−2.77 − 2.73i)10-s − 1.01i·11-s + (6.92 − 0.112i)12-s − 3.44·13-s + 3.37i·15-s + (−15.9 + 0.518i)16-s − 26.4·17-s + (−4.27 − 4.20i)18-s + 33.7i·19-s + ⋯
L(s)  = 1  + (0.712 + 0.701i)2-s − 0.577i·3-s + (0.0161 + 0.999i)4-s − 0.389·5-s + (0.404 − 0.411i)6-s + (−0.689 + 0.724i)8-s − 0.333·9-s + (−0.277 − 0.273i)10-s − 0.0923i·11-s + (0.577 − 0.00934i)12-s − 0.264·13-s + 0.224i·15-s + (−0.999 + 0.0323i)16-s − 1.55·17-s + (−0.237 − 0.233i)18-s + 1.77i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0161i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.999 + 0.0161i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.999 + 0.0161i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8605300150\)
\(L(\frac12)\) \(\approx\) \(0.8605300150\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.42 - 1.40i)T \)
3 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + 1.94T + 25T^{2} \)
11 \( 1 + 1.01iT - 121T^{2} \)
13 \( 1 + 3.44T + 169T^{2} \)
17 \( 1 + 26.4T + 289T^{2} \)
19 \( 1 - 33.7iT - 361T^{2} \)
23 \( 1 - 2.98iT - 529T^{2} \)
29 \( 1 - 30.3T + 841T^{2} \)
31 \( 1 - 0.713iT - 961T^{2} \)
37 \( 1 + 60.0T + 1.36e3T^{2} \)
41 \( 1 + 2.67T + 1.68e3T^{2} \)
43 \( 1 - 6.69iT - 1.84e3T^{2} \)
47 \( 1 + 24.3iT - 2.20e3T^{2} \)
53 \( 1 + 65.8T + 2.80e3T^{2} \)
59 \( 1 + 33.7iT - 3.48e3T^{2} \)
61 \( 1 - 81.7T + 3.72e3T^{2} \)
67 \( 1 - 62.6iT - 4.48e3T^{2} \)
71 \( 1 - 132. iT - 5.04e3T^{2} \)
73 \( 1 + 100.T + 5.32e3T^{2} \)
79 \( 1 + 97.5iT - 6.24e3T^{2} \)
83 \( 1 - 87.9iT - 6.88e3T^{2} \)
89 \( 1 - 110.T + 7.92e3T^{2} \)
97 \( 1 + 35.6T + 9.40e3T^{2} \)
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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

−11.27196034158300379566028607521, −10.06400545391604153463805517718, −8.714238643213089516342697798354, −8.149278010725226898242148350785, −7.21969628025680431878412408844, −6.46149766395645430561980236929, −5.55525355515797707524716167826, −4.41776262681442795172735949160, −3.43401436669275876273563992254, −2.05578523744090128327793376590, 0.23532129899127006152684518778, 2.18909240291951516179442889313, 3.26896146099687636378339636414, 4.45100179348061967404830700532, 4.92576830325833043390801182582, 6.25815335649635985518945036622, 7.12426634725354967174410990116, 8.635456328082358669548650271334, 9.306001744614207406770781131869, 10.28400325750271003351225162197

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