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Properties

Label 2-588-4.3-c2-0-55
Degree $2$
Conductor $588$
Sign $-0.0345 + 0.999i$
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  + (0.0345 − 1.99i)2-s + 1.73i·3-s + (−3.99 − 0.138i)4-s + 6.29·5-s + (3.46 + 0.0599i)6-s + (−0.415 + 7.98i)8-s − 2.99·9-s + (0.217 − 12.5i)10-s − 16.5i·11-s + (0.239 − 6.92i)12-s − 14.1·13-s + 10.8i·15-s + (15.9 + 1.10i)16-s + 20.4·17-s + (−0.103 + 5.99i)18-s + 6.39i·19-s + ⋯
L(s)  = 1  + (0.0172 − 0.999i)2-s + 0.577i·3-s + (−0.999 − 0.0345i)4-s + 1.25·5-s + (0.577 + 0.00998i)6-s + (−0.0518 + 0.998i)8-s − 0.333·9-s + (0.0217 − 1.25i)10-s − 1.50i·11-s + (0.0199 − 0.577i)12-s − 1.08·13-s + 0.726i·15-s + (0.997 + 0.0691i)16-s + 1.20·17-s + (−0.00576 + 0.333i)18-s + 0.336i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0345 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0345 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0345 + 0.999i$
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: \((2,\ 588,\ (\ :1),\ -0.0345 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.881413656\)
\(L(\frac12)\) \(\approx\) \(1.881413656\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0345 + 1.99i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 6.29T + 25T^{2} \)
11 \( 1 + 16.5iT - 121T^{2} \)
13 \( 1 + 14.1T + 169T^{2} \)
17 \( 1 - 20.4T + 289T^{2} \)
19 \( 1 - 6.39iT - 361T^{2} \)
23 \( 1 + 29.0iT - 529T^{2} \)
29 \( 1 - 54.6T + 841T^{2} \)
31 \( 1 + 14.7iT - 961T^{2} \)
37 \( 1 - 26.8T + 1.36e3T^{2} \)
41 \( 1 - 33.5T + 1.68e3T^{2} \)
43 \( 1 + 3.83iT - 1.84e3T^{2} \)
47 \( 1 + 27.4iT - 2.20e3T^{2} \)
53 \( 1 + 21.7T + 2.80e3T^{2} \)
59 \( 1 + 54.1iT - 3.48e3T^{2} \)
61 \( 1 - 35.7T + 3.72e3T^{2} \)
67 \( 1 + 95.4iT - 4.48e3T^{2} \)
71 \( 1 - 55.6iT - 5.04e3T^{2} \)
73 \( 1 + 130.T + 5.32e3T^{2} \)
79 \( 1 + 15.6iT - 6.24e3T^{2} \)
83 \( 1 - 49.4iT - 6.88e3T^{2} \)
89 \( 1 + 39.2T + 7.92e3T^{2} \)
97 \( 1 - 171.T + 9.40e3T^{2} \)
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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

−10.14797020088554928844577891433, −9.767113529651062317009453723768, −8.806279698620640452338040224813, −8.015576663659952979918023866775, −6.22180429436421282709811621455, −5.50304560397425304734393205993, −4.56835565961379705018391046856, −3.22020991167486311319904012894, −2.39405030654594287855003011811, −0.793056235069887827694942430241, 1.34998715799338370803034262777, 2.73642835302585744535228899167, 4.56053945147024004651010761735, 5.38504078629254047506567809086, 6.22588888373521036045731082210, 7.19866489996950506931248645529, 7.69310433840910500909099866114, 8.968288333554803022102982667986, 9.869518621932494457589941757042, 10.08719410507932392855246847313

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