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Properties

Label 2-560-35.17-c1-0-2
Degree $2$
Conductor $560$
Sign $0.0932 - 0.995i$
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 − 0.133i)3-s + (−1.86 − 1.23i)5-s + (0.866 + 2.5i)7-s + (−2.36 + 1.36i)9-s + (−1.36 + 2.36i)11-s + (2 + 2i)13-s + (−1.09 − 0.366i)15-s + (1 + 3.73i)17-s + (0.366 + 0.633i)19-s + (0.767 + 1.13i)21-s + (0.133 + 0.0358i)23-s + (1.96 + 4.59i)25-s + (−2.09 + 2.09i)27-s − 3i·29-s + (6.46 + 3.73i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.0773i)3-s + (−0.834 − 0.550i)5-s + (0.327 + 0.944i)7-s + (−0.788 + 0.455i)9-s + (−0.411 + 0.713i)11-s + (0.554 + 0.554i)13-s + (−0.283 − 0.0945i)15-s + (0.242 + 0.905i)17-s + (0.0839 + 0.145i)19-s + (0.167 + 0.247i)21-s + (0.0279 + 0.00748i)23-s + (0.392 + 0.919i)25-s + (−0.403 + 0.403i)27-s − 0.557i·29-s + (1.16 + 0.670i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.0932 - 0.995i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.0932 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807883 + 0.735780i\)
\(L(\frac12)\) \(\approx\) \(0.807883 + 0.735780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 + (-0.866 - 2.5i)T \)
good3 \( 1 + (-0.5 + 0.133i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (-1 - 3.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.366 - 0.633i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.133 - 0.0358i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-6.46 - 3.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 4.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.46iT - 41T^{2} \)
43 \( 1 + (2.83 - 2.83i)T - 43iT^{2} \)
47 \( 1 + (8.83 + 2.36i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.83 + 6.83i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.09 - 7.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.33 + 0.767i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 2.86i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.26T + 71T^{2} \)
73 \( 1 + (12.9 - 3.46i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.83 + 1.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.09 + 2.09i)T + 83iT^{2} \)
89 \( 1 + (-0.330 - 0.571i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.92 + 5.92i)T - 97iT^{2} \)
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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

−11.24275208463552117778675068223, −10.03682432347332039799618075145, −8.939167981926123511137726079905, −8.304092586772914884447067114533, −7.76426792423739701654450713441, −6.37408812913633825304646863612, −5.30524412108331214043266418814, −4.42770829381474179927775566291, −3.11902273669145309725029451680, −1.80159770493756240837835762084, 0.61201967691215579748283958416, 2.92532094509204936491442098845, 3.57363437185351871256187026249, 4.79477668246437205140889889478, 6.06045634547949634619924648105, 7.08132298194519879940526995035, 7.996396279769337881828196197100, 8.520612227261332822191291905175, 9.778858117447348769745184856362, 10.72723422651139804998018431115

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