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Properties

Label 1-588-588.47-r1-0-0
Degree $1$
Conductor $588$
Sign $-0.754 - 0.656i$
Analytic cond. $63.1893$
Root an. cond. $63.1893$
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  + (0.955 − 0.294i)5-s + (0.0747 − 0.997i)11-s + (0.900 − 0.433i)13-s + (−0.988 + 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.623 − 0.781i)29-s + (−0.5 + 0.866i)31-s + (0.365 − 0.930i)37-s + (−0.222 + 0.974i)41-s + (0.222 + 0.974i)43-s + (−0.826 − 0.563i)47-s + (−0.365 − 0.930i)53-s + (−0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)5-s + (0.0747 − 0.997i)11-s + (0.900 − 0.433i)13-s + (−0.988 + 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.623 − 0.781i)29-s + (−0.5 + 0.866i)31-s + (0.365 − 0.930i)37-s + (−0.222 + 0.974i)41-s + (0.222 + 0.974i)43-s + (−0.826 − 0.563i)47-s + (−0.365 − 0.930i)53-s + (−0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.754 - 0.656i$
Analytic conductor: \(63.1893\)
Root analytic conductor: \(63.1893\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 588,\ (1:\ ),\ -0.754 - 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5081534777 - 1.358380668i\)
\(L(\frac12)\) \(\approx\) \(0.5081534777 - 1.358380668i\)
\(L(1)\) \(\approx\) \(1.069762857 - 0.3053924260i\)
\(L(1)\) \(\approx\) \(1.069762857 - 0.3053924260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.365 - 0.930i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.365 - 0.930i)T \)
59 \( 1 + (-0.955 - 0.294i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
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

−23.22201567727425247796338797057, −22.30264951607910597273515255514, −21.77144426968205078580025632057, −20.60269802974356273694990028153, −20.31221874889646886313643121889, −18.88079523776257922499936975247, −18.2775145619769237252773063955, −17.4948185625756072205313972321, −16.7176334259169180296696184347, −15.67865129808280502878961241317, −14.78662778572167872617578495280, −13.96134018167168510090024785415, −13.19980192240992551289991925557, −12.32928191328459670533542518344, −11.19811858254608880360063205990, −10.38018245917775500343478949515, −9.523232043319414209707765748114, −8.74527095316317678353540659889, −7.510942158981718676028600414837, −6.51628615875878170752902371876, −5.84812407563682355518646251091, −4.637287052583826770656585232903, −3.6380936454416847966053778712, −2.20320897425153144046227685626, −1.58940944953318256754473203489, 0.31957203417372558257268275567, 1.572089968897326351353660322826, 2.63990399136381260469783699364, 3.83616506918212496487208522073, 4.99200016399691260441951507060, 6.026630617482774181961908357273, 6.536424770590833276680699048403, 8.06858207319210441796106487443, 8.802400427296799242366690812167, 9.61301566004454312429611099868, 10.751690454556410127798637236185, 11.29179163568273037524920024779, 12.69518943672026239439563510698, 13.35513689302677225981125889294, 13.971886744184486033556643662616, 15.06004509556203430684618607172, 16.06856096528381572652942872874, 16.72233496369745877962869003771, 17.84277094657498319944232376513, 18.15813845450101502328073351387, 19.43110442633142773153666275475, 20.14464448939031325199523843629, 21.17053468856520087527838441159, 21.67794968722340550773308799343, 22.48736554824638692123500226444

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