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Properties

Label 2-560-35.33-c1-0-15
Degree $2$
Conductor $560$
Sign $0.995 - 0.0932i$
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  + (1.86 + 0.5i)3-s + (1.86 − 1.23i)5-s + (2.5 + 0.866i)7-s + (0.633 + 0.366i)9-s + (0.366 + 0.633i)11-s + (−2 + 2i)13-s + (4.09 − 1.36i)15-s + (0.267 − i)17-s + (1.36 − 2.36i)19-s + (4.23 + 2.86i)21-s + (−6.96 + 1.86i)23-s + (1.96 − 4.59i)25-s + (−3.09 − 3.09i)27-s + 3i·29-s + (−0.464 + 0.267i)31-s + ⋯
L(s)  = 1  + (1.07 + 0.288i)3-s + (0.834 − 0.550i)5-s + (0.944 + 0.327i)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 + (0.0649 − 0.242i)17-s + (0.313 − 0.542i)19-s + (0.923 + 0.625i)21-s + (−1.45 + 0.389i)23-s + (0.392 − 0.919i)25-s + (−0.596 − 0.596i)27-s + 0.557i·29-s + (−0.0833 + 0.0481i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.995 - 0.0932i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.995 - 0.0932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42048 + 0.113059i\)
\(L(\frac12)\) \(\approx\) \(2.42048 + 0.113059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good3 \( 1 + (-1.86 - 0.5i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.366 - 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (-0.267 + i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.36 + 2.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.96 - 1.86i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (0.464 - 0.267i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.26 + 4.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.464iT - 41T^{2} \)
43 \( 1 + (-5.83 - 5.83i)T + 43iT^{2} \)
47 \( 1 + (0.633 - 0.169i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.83 - 6.83i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (1.13 + 0.303i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.73T + 71T^{2} \)
73 \( 1 + (-3.46 - 0.928i)T + (63.2 + 36.5i)T^{2} \)
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

−10.61736277797178736582868960099, −9.458499261095032836911439339387, −9.224938868269836061399974179933, −8.266957486075104034554208588391, −7.48071556414062015574682885705, −6.08156285394035009559105904264, −5.06807857966314492239820865664, −4.14982750212631766048856426621, −2.66028460896628459813390347535, −1.74302988403643170676739164818, 1.73320803368637708840705051486, 2.63841316684878581418881363766, 3.82859304990681205797880679444, 5.23405700479896909505779315334, 6.21199058081073731396058415306, 7.47540805662847755451117666927, 8.004213607318069454388351241156, 8.888214348022507929830353531787, 9.926109155312349395744118873160, 10.53235201770152034892113155927

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