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Properties

Label 2-579-579.362-c0-0-0
Degree $2$
Conductor $579$
Sign $0.938 - 0.345i$
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 + (0.785 − 0.785i)7-s + (−0.707 + 0.707i)9-s + (0.707 + 0.707i)12-s + (−1.83 − 0.555i)13-s + (0.707 − 0.707i)16-s + (−1.26 + 0.124i)19-s + (1.02 + 0.425i)21-s + (−0.980 + 0.195i)25-s + (−0.923 − 0.382i)27-s + (0.425 − 1.02i)28-s + (0.360 + 1.81i)31-s + (−0.382 + 0.923i)36-s + (0.831 + 1.55i)37-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (0.923 − 0.382i)4-s + (0.785 − 0.785i)7-s + (−0.707 + 0.707i)9-s + (0.707 + 0.707i)12-s + (−1.83 − 0.555i)13-s + (0.707 − 0.707i)16-s + (−1.26 + 0.124i)19-s + (1.02 + 0.425i)21-s + (−0.980 + 0.195i)25-s + (−0.923 − 0.382i)27-s + (0.425 − 1.02i)28-s + (0.360 + 1.81i)31-s + (−0.382 + 0.923i)36-s + (0.831 + 1.55i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193236339\)
\(L(\frac12)\) \(\approx\) \(1.193236339\)
\(L(1)\) 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 \( 1 + (0.980 - 0.195i)T^{2} \)
7 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
11 \( 1 + (-0.195 + 0.980i)T^{2} \)
13 \( 1 + (1.83 + 0.555i)T + (0.831 + 0.555i)T^{2} \)
17 \( 1 + (0.980 + 0.195i)T^{2} \)
19 \( 1 + (1.26 - 0.124i)T + (0.980 - 0.195i)T^{2} \)
23 \( 1 + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (0.555 + 0.831i)T^{2} \)
31 \( 1 + (-0.360 - 1.81i)T + (-0.923 + 0.382i)T^{2} \)
37 \( 1 + (-0.831 - 1.55i)T + (-0.555 + 0.831i)T^{2} \)
41 \( 1 + (-0.195 + 0.980i)T^{2} \)
43 \( 1 + (0.275 + 0.275i)T + iT^{2} \)
47 \( 1 + (0.831 + 0.555i)T^{2} \)
53 \( 1 + (-0.831 - 0.555i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.195 - 0.0192i)T + (0.980 + 0.195i)T^{2} \)
67 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.980 + 0.195i)T^{2} \)
73 \( 1 + (-0.938 + 1.75i)T + (-0.555 - 0.831i)T^{2} \)
79 \( 1 + (-0.598 - 0.728i)T + (-0.195 + 0.980i)T^{2} \)
83 \( 1 + (-0.382 - 0.923i)T^{2} \)
89 \( 1 + (-0.831 + 0.555i)T^{2} \)
97 \( 1 + (-1.38 - 0.275i)T + (0.923 + 0.382i)T^{2} \)
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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

−10.67207631105769072877135369155, −10.31495380698388634952432455110, −9.502087501025703921029005323812, −8.163800258876048372032151227860, −7.58976366365501653661722212230, −6.49879250002570248333756845334, −5.18989724564568915021220570354, −4.54280033678521130184425017380, −3.14008141214671876647346594448, −2.00612519443209139350884772468, 2.21027825867868094838694125680, 2.36062547143334541703062016245, 4.16018230307524136011547803808, 5.61408587762114590551532711463, 6.50788422789041034887648099902, 7.46954149467975515084551961290, 7.972145143595714235980692605146, 8.925591323397484351612560545858, 9.946229917013414090961668171627, 11.33513345773501539446855280256

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