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Properties

Label 2-579-579.428-c0-0-0
Degree $2$
Conductor $579$
Sign $0.500 - 0.865i$
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 + (0.275 + 0.275i)7-s + (0.707 + 0.707i)9-s + (−0.707 + 0.707i)12-s + (−1.98 − 0.195i)13-s + (−0.707 − 0.707i)16-s + (0.902 − 1.68i)19-s + (0.149 + 0.360i)21-s + (0.555 + 0.831i)25-s + (0.382 + 0.923i)27-s + (−0.360 + 0.149i)28-s + (0.636 − 0.425i)31-s + (−0.923 + 0.382i)36-s + (0.980 − 0.804i)37-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)3-s + (−0.382 + 0.923i)4-s + (0.275 + 0.275i)7-s + (0.707 + 0.707i)9-s + (−0.707 + 0.707i)12-s + (−1.98 − 0.195i)13-s + (−0.707 − 0.707i)16-s + (0.902 − 1.68i)19-s + (0.149 + 0.360i)21-s + (0.555 + 0.831i)25-s + (0.382 + 0.923i)27-s + (−0.360 + 0.149i)28-s + (0.636 − 0.425i)31-s + (−0.923 + 0.382i)36-s + (0.980 − 0.804i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ 0.500 - 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.097218280\)
\(L(\frac12)\) \(\approx\) \(1.097218280\)
\(L(1)\) 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 \( 1 + (-0.555 - 0.831i)T^{2} \)
7 \( 1 + (-0.275 - 0.275i)T + iT^{2} \)
11 \( 1 + (0.831 + 0.555i)T^{2} \)
13 \( 1 + (1.98 + 0.195i)T + (0.980 + 0.195i)T^{2} \)
17 \( 1 + (-0.555 + 0.831i)T^{2} \)
19 \( 1 + (-0.902 + 1.68i)T + (-0.555 - 0.831i)T^{2} \)
23 \( 1 + (0.382 - 0.923i)T^{2} \)
29 \( 1 + (-0.195 - 0.980i)T^{2} \)
31 \( 1 + (-0.636 + 0.425i)T + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.980 + 0.804i)T + (0.195 - 0.980i)T^{2} \)
41 \( 1 + (0.831 + 0.555i)T^{2} \)
43 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
47 \( 1 + (0.980 + 0.195i)T^{2} \)
53 \( 1 + (-0.980 - 0.195i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.831 + 1.55i)T + (-0.555 + 0.831i)T^{2} \)
67 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (0.555 + 0.831i)T^{2} \)
73 \( 1 + (-0.728 - 0.598i)T + (0.195 + 0.980i)T^{2} \)
79 \( 1 + (1.47 + 0.448i)T + (0.831 + 0.555i)T^{2} \)
83 \( 1 + (-0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.980 + 0.195i)T^{2} \)
97 \( 1 + (-0.785 + 1.17i)T + (-0.382 - 0.923i)T^{2} \)
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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

−11.15091626094203980257716087686, −9.760901475235132932047429370359, −9.388565781945554017274359867753, −8.463421032699626680151285272099, −7.58546213355397238565580636375, −7.07271978519665541269562758753, −5.05378635130402723723286582633, −4.57385390907522152916764662229, −3.15663414733600641272031304465, −2.47465184985798324585479870430, 1.47213214605567126917006251392, 2.74496134823515485424242581257, 4.22254023163715847366769220372, 5.10251200395587312231064995253, 6.34233125879643264415182471535, 7.34475902972355214474966584384, 8.102992607089629385281885087839, 9.129919712550333980905627195116, 9.977697013647465102776627323652, 10.30165864489468246401694277751

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