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Properties

Label 2-560-35.4-c1-0-3
Degree $2$
Conductor $560$
Sign $0.330 - 0.943i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (0.133 + 2.23i)5-s + (1.73 + 2i)7-s + (−1.5 − 2.59i)9-s + (1.5 − 2.59i)11-s + 5i·13-s + (1.73 + i)17-s + (2.5 + 4.33i)19-s + (−6.06 + 3.5i)23-s + (−4.96 + 0.598i)25-s + 4·29-s + (−1 + 1.73i)31-s + (−4.23 + 4.13i)35-s + (0.866 − 0.5i)37-s + 3·41-s + 2i·43-s + ⋯
L(s)  = 1  + (0.0599 + 0.998i)5-s + (0.654 + 0.755i)7-s + (−0.5 − 0.866i)9-s + (0.452 − 0.783i)11-s + 1.38i·13-s + (0.420 + 0.242i)17-s + (0.573 + 0.993i)19-s + (−1.26 + 0.729i)23-s + (−0.992 + 0.119i)25-s + 0.742·29-s + (−0.179 + 0.311i)31-s + (−0.715 + 0.698i)35-s + (0.142 − 0.0821i)37-s + 0.468·41-s + 0.304i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20268 + 0.853082i\)
\(L(\frac12)\) \(\approx\) \(1.20268 + 0.853082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.133 - 2.23i)T \)
7 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-6.06 + 3.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.79 - 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (13.8 + 8i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−11.12084112947036548749192000816, −10.01659348280096301328209811637, −9.171719699090685490862772077543, −8.359623364672106616537035465382, −7.32627038122318878141957151154, −6.17563815887789357044969176875, −5.77046415430889980623262016103, −4.08641743454540912391561741134, −3.15130592821623598128012225223, −1.77292869520538079212190014310, 0.909388870489807595718687289013, 2.44087868296226965242062633353, 4.12106222277381466252963932499, 4.95366397005094649059406455067, 5.72574568996022039296566000170, 7.25400471037142305563566122526, 7.969902570913491715807062164465, 8.660404202337105468623010409114, 9.843912950169102158092732767051, 10.49213872839324174108449622629

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