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Properties

Label 2-579-579.377-c0-0-0
Degree $2$
Conductor $579$
Sign $-0.555 + 0.831i$
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  − i·3-s − i·4-s − 1.41·7-s − 9-s − 12-s + (1.70 − 0.707i)13-s − 16-s + (−0.292 − 0.707i)19-s + 1.41i·21-s + (−0.707 + 0.707i)25-s + i·27-s + 1.41i·28-s + i·36-s + (1.70 − 0.707i)37-s + (−0.707 − 1.70i)39-s + ⋯
L(s)  = 1  − i·3-s − i·4-s − 1.41·7-s − 9-s − 12-s + (1.70 − 0.707i)13-s − 16-s + (−0.292 − 0.707i)19-s + 1.41i·21-s + (−0.707 + 0.707i)25-s + i·27-s + 1.41i·28-s + i·36-s + (1.70 − 0.707i)37-s + (−0.707 − 1.70i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ -0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7533616352\)
\(L(\frac12)\) \(\approx\) \(0.7533616352\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
193 \( 1 + iT \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
17 \( 1 + (0.707 + 0.707i)T^{2} \)
19 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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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

−10.82939296645155943436779078849, −9.631367035271602421193319729223, −9.011143208275288189815943524233, −7.907813059550447884630181289659, −6.74556035286395951908503382489, −6.16049303606709725030398261028, −5.52568849738607541054629704852, −3.75455629088721001147603203742, −2.51287926599909745018012277628, −0.927301316359729147160371204148, 2.72731989363962471838607468974, 3.74459025598583700073774368365, 4.21443321376005432534623740008, 5.94693869499295963214687910180, 6.50829056996623940247723754289, 7.907757493115962320750593749748, 8.758472035653471157597464662617, 9.412322776982579213766682695033, 10.27792849413901919466873272669, 11.23237918449931720925883875412

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