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Properties

Label 2-578-17.13-c1-0-16
Degree $2$
Conductor $578$
Sign $-0.995 + 0.0929i$
Analytic cond. $4.61535$
Root an. cond. $2.14833$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (−1.41 − 1.41i)3-s − 4-s + (−1.41 + 1.41i)6-s + (2.82 − 2.82i)7-s + i·8-s + 1.00i·9-s + (4.24 − 4.24i)11-s + (1.41 + 1.41i)12-s − 2·13-s + (−2.82 − 2.82i)14-s + 16-s + 1.00·18-s + 4i·19-s − 8.00·21-s + (−4.24 − 4.24i)22-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.816 − 0.816i)3-s − 0.5·4-s + (−0.577 + 0.577i)6-s + (1.06 − 1.06i)7-s + 0.353i·8-s + 0.333i·9-s + (1.27 − 1.27i)11-s + (0.408 + 0.408i)12-s − 0.554·13-s + (−0.755 − 0.755i)14-s + 0.250·16-s + 0.235·18-s + 0.917i·19-s − 1.74·21-s + (−0.904 − 0.904i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $-0.995 + 0.0929i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ -0.995 + 0.0929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0507163 - 1.08917i\)
\(L(\frac12)\) \(\approx\) \(0.0507163 - 1.08917i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
17 \( 1 \)
good3 \( 1 + (1.41 + 1.41i)T + 3iT^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 + (-2.82 + 2.82i)T - 7iT^{2} \)
11 \( 1 + (-4.24 + 4.24i)T - 11iT^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + (2.82 + 2.82i)T + 31iT^{2} \)
37 \( 1 + (2.82 + 2.82i)T + 37iT^{2} \)
41 \( 1 + (4.24 - 4.24i)T - 41iT^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-2.82 + 2.82i)T - 61iT^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (1.41 + 1.41i)T + 73iT^{2} \)
79 \( 1 + (-5.65 + 5.65i)T - 79iT^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (9.89 + 9.89i)T + 97iT^{2} \)
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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

−10.65405623808610817601939765748, −9.595967587414894147131804856120, −8.450428505869644813097565012563, −7.64136731893460721190314009316, −6.60397760351583075174642803899, −5.74414656046482943532479072670, −4.50835948542159685202479396402, −3.55479797676155568159522925465, −1.67379944054990396075516303626, −0.74309729560089933688311215274, 1.95450376278110909512061469726, 3.98322818851447371954954792219, 5.07368114659910535256393513928, 5.23083106505850872460240477625, 6.60557397015475694808242274005, 7.42577533131738443743673828245, 8.667587891418158906031065554869, 9.313532388107732960994718405197, 10.15401008523471904547935739425, 11.24848526346286523861087809882

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