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Properties

Label 2-579-579.260-c0-0-0
Degree $2$
Conductor $579$
Sign $-0.712 - 0.701i$
Analytic cond. $0.288958$
Root an. cond. $0.537548$
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  + (−0.923 + 0.382i)3-s + (0.382 + 0.923i)4-s + (−1.38 + 1.38i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 0.707i)12-s + (−1.19 − 0.980i)13-s + (−0.707 + 0.707i)16-s + (−0.273 + 0.902i)19-s + (0.750 − 1.81i)21-s + (0.831 + 0.555i)25-s + (−0.382 + 0.923i)27-s + (−1.81 − 0.750i)28-s + (0.425 − 0.636i)31-s + (0.923 + 0.382i)36-s + (0.195 + 1.98i)37-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (0.382 + 0.923i)4-s + (−1.38 + 1.38i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 0.707i)12-s + (−1.19 − 0.980i)13-s + (−0.707 + 0.707i)16-s + (−0.273 + 0.902i)19-s + (0.750 − 1.81i)21-s + (0.831 + 0.555i)25-s + (−0.382 + 0.923i)27-s + (−1.81 − 0.750i)28-s + (0.425 − 0.636i)31-s + (0.923 + 0.382i)36-s + (0.195 + 1.98i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (260, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5169870260\)
\(L(\frac12)\) \(\approx\) \(0.5169870260\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 - 0.382i)T \)
193 \( 1 + (-0.382 - 0.923i)T \)
good2 \( 1 + (-0.382 - 0.923i)T^{2} \)
5 \( 1 + (-0.831 - 0.555i)T^{2} \)
7 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
11 \( 1 + (-0.555 - 0.831i)T^{2} \)
13 \( 1 + (1.19 + 0.980i)T + (0.195 + 0.980i)T^{2} \)
17 \( 1 + (-0.831 + 0.555i)T^{2} \)
19 \( 1 + (0.273 - 0.902i)T + (-0.831 - 0.555i)T^{2} \)
23 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.980 + 0.195i)T^{2} \)
31 \( 1 + (-0.425 + 0.636i)T + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.195 - 1.98i)T + (-0.980 + 0.195i)T^{2} \)
41 \( 1 + (-0.555 - 0.831i)T^{2} \)
43 \( 1 + (-0.785 - 0.785i)T + iT^{2} \)
47 \( 1 + (0.195 + 0.980i)T^{2} \)
53 \( 1 + (-0.195 - 0.980i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.555 - 1.83i)T + (-0.831 + 0.555i)T^{2} \)
67 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
71 \( 1 + (0.831 + 0.555i)T^{2} \)
73 \( 1 + (-0.0569 + 0.577i)T + (-0.980 - 0.195i)T^{2} \)
79 \( 1 + (-0.0924 + 0.172i)T + (-0.555 - 0.831i)T^{2} \)
83 \( 1 + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.195 + 0.980i)T^{2} \)
97 \( 1 + (-1.17 + 0.785i)T + (0.382 - 0.923i)T^{2} \)
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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

−11.48743775168665510470266722301, −10.28255772013177823573366214847, −9.670485191955309152999294035430, −8.720299846366996678334161483089, −7.61429478776640967648315370638, −6.55947025108208249793057494401, −5.91717873181657465373548366769, −4.86636103227121054873200451114, −3.44499658068717186220156886001, −2.61428054061799489162970826165, 0.65141836869973451084679036472, 2.38204597958068103019312494731, 4.16319431683690975575403466569, 5.09124093147778548427810330816, 6.30630703922784714687427532521, 6.90746320370926791955195623591, 7.32050167932859024639235672147, 9.231820960145120662974536870916, 9.963241943442798319668852401594, 10.60256839336485605372766009203

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