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Properties

Label 1-588-588.23-r0-0-0
Degree $1$
Conductor $588$
Sign $0.871 - 0.490i$
Analytic cond. $2.73066$
Root an. cond. $2.73066$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0747 − 0.997i)5-s + (0.365 + 0.930i)11-s + (0.623 − 0.781i)13-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (0.222 − 0.974i)29-s + (0.5 − 0.866i)31-s + (0.955 − 0.294i)37-s + (0.900 + 0.433i)41-s + (0.900 − 0.433i)43-s + (−0.988 − 0.149i)47-s + (−0.955 − 0.294i)53-s + (0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)5-s + (0.365 + 0.930i)11-s + (0.623 − 0.781i)13-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (0.222 − 0.974i)29-s + (0.5 − 0.866i)31-s + (0.955 − 0.294i)37-s + (0.900 + 0.433i)41-s + (0.900 − 0.433i)43-s + (−0.988 − 0.149i)47-s + (−0.955 − 0.294i)53-s + (0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(2.73066\)
Root analytic conductor: \(2.73066\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 588,\ (0:\ ),\ 0.871 - 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417349867 - 0.3716729040i\)
\(L(\frac12)\) \(\approx\) \(1.417349867 - 0.3716729040i\)
\(L(1)\) \(\approx\) \(1.132180437 - 0.1586285170i\)
\(L(1)\) \(\approx\) \(1.132180437 - 0.1586285170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.22827938420948872356914817136, −22.30106520356075029183421107161, −21.71742111245170038349024569067, −20.86431625835125515773191449018, −19.75196116463060848899554542189, −19.00382086494126038756695720712, −18.32904322059904164974353093269, −17.570502718389385460226549268877, −16.20936956212380296888947557832, −15.97449989566177478654208982292, −14.43882467589935972078969097230, −14.23305250878348763568441915594, −13.23378286720696833670317778818, −11.90503382131731336416906749974, −11.30791523189500856511847944850, −10.4933576478481341794031762350, −9.45503771511227095529388219836, −8.54343368842362200707158544184, −7.445656178974453452719870396965, −6.58809351971322472832377802817, −5.8263769926973736994065227275, −4.48743717260249478934041240537, −3.38512263617308550331175066434, −2.63072316750490490165014455401, −1.110666401889176061385999006628, 0.984829110680570416027159110335, 2.006063819467474532190279395464, 3.58110833154723730666791759731, 4.35768757614780427588003719612, 5.50962718012470330662202636355, 6.21103056286087249257681569549, 7.83660541445680778346590734735, 8.06839902853521564253261527588, 9.50504434751128246764042126541, 9.92621204644912870083888622451, 11.25782728133658004635486810938, 12.20733927120213790260638938753, 12.7759308246514948106293240048, 13.70437126077537934494286096229, 14.73954347966823442317907296746, 15.62837510626831709849316191973, 16.37650912453657598738763774747, 17.30604216277475906697551235937, 17.89710242657566092997379080982, 19.06601864187000896771293226118, 19.90169155307501711157283998350, 20.63482739815756768402306062630, 21.19373442618378760728032416640, 22.397769354903132965853361497314, 23.13809187228364474271855552335

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