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Properties

Label 2-579-579.389-c0-0-0
Degree $2$
Conductor $579$
Sign $0.936 + 0.350i$
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  + (−0.707 + 0.707i)3-s + (0.707 − 0.707i)4-s − 0.765i·7-s − 1.00i·9-s + 1.00i·12-s + (0.617 − 0.923i)13-s − 1.00i·16-s + (0.382 + 1.92i)19-s + (0.541 + 0.541i)21-s + (−0.923 + 0.382i)25-s + (0.707 + 0.707i)27-s + (−0.541 − 0.541i)28-s + (1.30 − 0.541i)31-s + (−0.707 − 0.707i)36-s + (−1.38 − 0.923i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.707 − 0.707i)4-s − 0.765i·7-s − 1.00i·9-s + 1.00i·12-s + (0.617 − 0.923i)13-s − 1.00i·16-s + (0.382 + 1.92i)19-s + (0.541 + 0.541i)21-s + (−0.923 + 0.382i)25-s + (0.707 + 0.707i)27-s + (−0.541 − 0.541i)28-s + (1.30 − 0.541i)31-s + (−0.707 − 0.707i)36-s + (−1.38 − 0.923i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ 0.936 + 0.350i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8591887373\)
\(L(\frac12)\) \(\approx\) \(0.8591887373\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
193 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.923 - 0.382i)T^{2} \)
7 \( 1 + 0.765iT - T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (0.923 + 0.382i)T^{2} \)
19 \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \)
41 \( 1 + (-0.923 - 0.382i)T^{2} \)
43 \( 1 - 1.84iT - T^{2} \)
47 \( 1 + (0.382 + 0.923i)T^{2} \)
53 \( 1 + (0.382 + 0.923i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.382 - 0.923i)T^{2} \)
97 \( 1 + (0.707 + 0.707i)T^{2} \)
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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

−10.65253177253885017516290231479, −10.25613327539658067925040448806, −9.567775093612490373250840030012, −8.136067219175168813121018945983, −7.17808773259324199318177262373, −5.99987342831053256747977903465, −5.65072563745387050715478823677, −4.30565578491865866604433573731, −3.26328444090209368783487250193, −1.29724726689964492860792659267, 1.85794122561233825586860779925, 2.92959789447541904208309200809, 4.50196742333211779539488076708, 5.68520931133271680274799415143, 6.65458960736908950803693255862, 7.13102378450156915298161998575, 8.305505008063984005331111539844, 8.966121019744763047236752066643, 10.39252729238521934296686478944, 11.29660633904027689505228405996

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