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Properties

Label 2-579-579.185-c0-0-0
Degree $2$
Conductor $579$
Sign $0.653 + 0.756i$
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 + (−0.168 − 0.555i)13-s + (0.707 + 0.707i)16-s + (−0.151 + 1.53i)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 − 0.444i)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 + (−0.168 − 0.555i)13-s + (0.707 + 0.707i)16-s + (−0.151 + 1.53i)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 − 0.444i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.653 + 0.756i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ 0.653 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117023626\)
\(L(\frac12)\) \(\approx\) \(1.117023626\)
\(L(1)\) 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 \( 1 + (0.168 + 0.555i)T + (-0.831 + 0.555i)T^{2} \)
17 \( 1 + (-0.980 + 0.195i)T^{2} \)
19 \( 1 + (0.151 - 1.53i)T + (-0.980 - 0.195i)T^{2} \)
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 \( 1 + (0.831 + 0.444i)T + (0.555 + 0.831i)T^{2} \)
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 \( 1 + (0.195 + 1.98i)T + (-0.980 + 0.195i)T^{2} \)
67 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
71 \( 1 + (0.980 + 0.195i)T^{2} \)
73 \( 1 + (0.172 - 0.0924i)T + (0.555 - 0.831i)T^{2} \)
79 \( 1 + (1.36 + 1.11i)T + (0.195 + 0.980i)T^{2} \)
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

−10.74127009919836613440545597164, −10.14896324551784165091859981989, −8.825657968990524181847410781684, −7.963549538783921277104210858340, −7.14851882433413638912147642428, −6.62236141925963446401248072249, −5.60015983535386887225549326769, −3.67910555656296836693685393571, −2.98152666404592363843685980755, −1.56142756919976868188868371924, 2.34148875325311238792463247072, 3.06965488896044187763369207869, 4.49841525502332942481930808319, 5.57167280303546846736267924476, 6.44922142637906038565225069640, 7.40834533323777937334464258602, 8.721620724252098461565045333940, 9.356053700171297330404133393199, 10.11394458640604484480666611988, 11.03616533230634314254705053846

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