Properties

Label 2-560-35.3-c1-0-17
Degree $2$
Conductor $560$
Sign $0.848 + 0.528i$
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  + (0.5 + 1.86i)3-s + (−0.133 − 2.23i)5-s + (−0.866 − 2.5i)7-s + (−0.633 + 0.366i)9-s + (0.366 − 0.633i)11-s + (2 − 2i)13-s + (4.09 − 1.36i)15-s + (1 − 0.267i)17-s + (−1.36 − 2.36i)19-s + (4.23 − 2.86i)21-s + (1.86 − 6.96i)23-s + (−4.96 + 0.598i)25-s + (3.09 + 3.09i)27-s + 3i·29-s + (−0.464 − 0.267i)31-s + ⋯
L(s)  = 1  + (0.288 + 1.07i)3-s + (−0.0599 − 0.998i)5-s + (−0.327 − 0.944i)7-s + (−0.211 + 0.122i)9-s + (0.110 − 0.191i)11-s + (0.554 − 0.554i)13-s + (1.05 − 0.352i)15-s + (0.242 − 0.0649i)17-s + (−0.313 − 0.542i)19-s + (0.923 − 0.625i)21-s + (0.389 − 1.45i)23-s + (−0.992 + 0.119i)25-s + (0.596 + 0.596i)27-s + 0.557i·29-s + (−0.0833 − 0.0481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.848 + 0.528i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.848 + 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46736 - 0.419564i\)
\(L(\frac12)\) \(\approx\) \(1.46736 - 0.419564i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 2.5i)T \)
good3 \( 1 + (-0.5 - 1.86i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.366 + 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (-1 + 0.267i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.36 + 2.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 + 6.96i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (0.464 + 0.267i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 1.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (-5.83 - 5.83i)T + 43iT^{2} \)
47 \( 1 + (0.169 - 0.633i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.83 + 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.09 + 1.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.33 - 4.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.303 - 1.13i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.73T + 71T^{2} \)
73 \( 1 + (-0.928 - 3.46i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.83 - 3.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.09 + 3.09i)T - 83iT^{2} \)
89 \( 1 + (8.33 + 14.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.92 + 7.92i)T + 97iT^{2} \)
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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

−10.52778810044323177770578778338, −9.826844302292207034364829750791, −8.979993543085423816835341089565, −8.322269558593354684035253872382, −7.15527135632301220956647719078, −5.95722975110971380557796720793, −4.70750490172076158066300270099, −4.17277695570369197425993102220, −3.08586664357771231855932281104, −0.919768922306661910640116325200, 1.73556471878754538975600940548, 2.71980241819848399931898225184, 3.91006254439122032185517464871, 5.66895033600833930747938904689, 6.41338532177598580278298978815, 7.24811577245422869171332855600, 7.980021468989728217613862527843, 9.025651123465120256463663574949, 9.905790996090266697897797915546, 10.98045839201117166889261629433

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