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Properties

Label 2-579-579.14-c0-0-0
Degree $2$
Conductor $579$
Sign $-0.0637 + 0.997i$
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  + (−0.382 − 0.923i)3-s + (−0.923 + 0.382i)4-s + (1.17 − 1.17i)7-s + (−0.707 + 0.707i)9-s + (0.707 + 0.707i)12-s + (−0.444 − 0.831i)13-s + (0.707 − 0.707i)16-s + (−1.53 − 1.26i)19-s + (−1.53 − 0.636i)21-s + (−0.195 − 0.980i)25-s + (0.923 + 0.382i)27-s + (−0.636 + 1.53i)28-s + (1.81 − 0.360i)31-s + (0.382 − 0.923i)36-s + (−0.555 + 1.83i)37-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (−0.923 + 0.382i)4-s + (1.17 − 1.17i)7-s + (−0.707 + 0.707i)9-s + (0.707 + 0.707i)12-s + (−0.444 − 0.831i)13-s + (0.707 − 0.707i)16-s + (−1.53 − 1.26i)19-s + (−1.53 − 0.636i)21-s + (−0.195 − 0.980i)25-s + (0.923 + 0.382i)27-s + (−0.636 + 1.53i)28-s + (1.81 − 0.360i)31-s + (0.382 − 0.923i)36-s + (−0.555 + 1.83i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0637 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0637 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $-0.0637 + 0.997i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ -0.0637 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6610748556\)
\(L(\frac12)\) \(\approx\) \(0.6610748556\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 + 0.923i)T \)
193 \( 1 + (0.923 - 0.382i)T \)
good2 \( 1 + (0.923 - 0.382i)T^{2} \)
5 \( 1 + (0.195 + 0.980i)T^{2} \)
7 \( 1 + (-1.17 + 1.17i)T - iT^{2} \)
11 \( 1 + (0.980 + 0.195i)T^{2} \)
13 \( 1 + (0.444 + 0.831i)T + (-0.555 + 0.831i)T^{2} \)
17 \( 1 + (0.195 - 0.980i)T^{2} \)
19 \( 1 + (1.53 + 1.26i)T + (0.195 + 0.980i)T^{2} \)
23 \( 1 + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (0.831 - 0.555i)T^{2} \)
31 \( 1 + (-1.81 + 0.360i)T + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.555 - 1.83i)T + (-0.831 - 0.555i)T^{2} \)
41 \( 1 + (0.980 + 0.195i)T^{2} \)
43 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
47 \( 1 + (-0.555 + 0.831i)T^{2} \)
53 \( 1 + (0.555 - 0.831i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.980 - 0.804i)T + (0.195 - 0.980i)T^{2} \)
67 \( 1 + (-0.382 + 0.0761i)T + (0.923 - 0.382i)T^{2} \)
71 \( 1 + (-0.195 - 0.980i)T^{2} \)
73 \( 1 + (-0.448 - 1.47i)T + (-0.831 + 0.555i)T^{2} \)
79 \( 1 + (-0.577 - 0.0569i)T + (0.980 + 0.195i)T^{2} \)
83 \( 1 + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.555 + 0.831i)T^{2} \)
97 \( 1 + (-0.275 + 1.38i)T + (-0.923 - 0.382i)T^{2} \)
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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

−10.76026591765328124498848409059, −10.00319547005833530417901112247, −8.491105220463292697115623086849, −8.113877624256935471288576818603, −7.29468383112713670514756819668, −6.25655394057975753934010011149, −4.83985455479542953987607797723, −4.43176987099314054071511815317, −2.66765771924745179308051197213, −0.895697912928697223669291142032, 2.03241737971293993293486947485, 3.89082497950114536236282401578, 4.69656675105380370902812330300, 5.45255795150681268373586316298, 6.22493214385851892428682806555, 7.962177414698799563062376394001, 8.838973210821108690221931685699, 9.239945423502890279778106053364, 10.33457508520922785124070417428, 10.97334424281360421289831065620

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