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Properties

Label 2-588-4.3-c2-0-47
Degree $2$
Conductor $588$
Sign $0.999 + 0.0432i$
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  + (−1.44 + 1.38i)2-s + 1.73i·3-s + (0.173 − 3.99i)4-s + 1.50·5-s + (−2.39 − 2.50i)6-s + (5.27 + 6.01i)8-s − 2.99·9-s + (−2.17 + 2.07i)10-s − 9.21i·11-s + (6.92 + 0.299i)12-s + 5.97·13-s + 2.60i·15-s + (−15.9 − 1.38i)16-s − 3.95·17-s + (4.33 − 4.14i)18-s − 3.59i·19-s + ⋯
L(s)  = 1  + (−0.722 + 0.691i)2-s + 0.577i·3-s + (0.0432 − 0.999i)4-s + 0.300·5-s + (−0.399 − 0.416i)6-s + (0.659 + 0.751i)8-s − 0.333·9-s + (−0.217 + 0.207i)10-s − 0.837i·11-s + (0.576 + 0.0249i)12-s + 0.459·13-s + 0.173i·15-s + (−0.996 − 0.0864i)16-s − 0.232·17-s + (0.240 − 0.230i)18-s − 0.189i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0432i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.999 + 0.0432i$
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: \((2,\ 588,\ (\ :1),\ 0.999 + 0.0432i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.114957044\)
\(L(\frac12)\) \(\approx\) \(1.114957044\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.44 - 1.38i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 1.50T + 25T^{2} \)
11 \( 1 + 9.21iT - 121T^{2} \)
13 \( 1 - 5.97T + 169T^{2} \)
17 \( 1 + 3.95T + 289T^{2} \)
19 \( 1 + 3.59iT - 361T^{2} \)
23 \( 1 + 21.6iT - 529T^{2} \)
29 \( 1 - 47.5T + 841T^{2} \)
31 \( 1 + 38.5iT - 961T^{2} \)
37 \( 1 + 52.4T + 1.36e3T^{2} \)
41 \( 1 - 64.9T + 1.68e3T^{2} \)
43 \( 1 + 16.8iT - 1.84e3T^{2} \)
47 \( 1 + 68.2iT - 2.20e3T^{2} \)
53 \( 1 + 18.1T + 2.80e3T^{2} \)
59 \( 1 + 49.0iT - 3.48e3T^{2} \)
61 \( 1 - 98.5T + 3.72e3T^{2} \)
67 \( 1 - 129. iT - 4.48e3T^{2} \)
71 \( 1 + 61.8iT - 5.04e3T^{2} \)
73 \( 1 - 95.7T + 5.32e3T^{2} \)
79 \( 1 - 13.1iT - 6.24e3T^{2} \)
83 \( 1 - 91.0iT - 6.88e3T^{2} \)
89 \( 1 - 90.7T + 7.92e3T^{2} \)
97 \( 1 - 7.05T + 9.40e3T^{2} \)
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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

−10.33939187550612714695641300273, −9.539771023756337095906733114410, −8.654498494453131728094143337420, −8.116432407259732098770104294910, −6.80141196334348458843192723914, −6.02386171507302526003504017933, −5.16237559027847341399744985624, −3.95997471181985903553813027245, −2.35863273165238314567743930637, −0.60670901459744255485356194599, 1.20715494885782849460994501300, 2.23839063692645708283834268552, 3.46428573840096318664287609749, 4.75367103279559468510502315787, 6.18945763555960871426343214117, 7.13271020314105876235982824046, 7.932391300189964548069936771214, 8.827549427141836921320243567167, 9.658457368408710144019385996009, 10.45138723379004704456028115958

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