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Properties

Label 2-560-560.453-c1-0-79
Degree $2$
Conductor $560$
Sign $-0.239 + 0.970i$
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  + (−1.32 + 0.481i)2-s + (0.689 − 0.398i)3-s + (1.53 − 1.28i)4-s + (1.03 − 1.98i)5-s + (−0.724 + 0.861i)6-s + (−1.14 − 2.38i)7-s + (−1.42 + 2.44i)8-s + (−1.18 + 2.04i)9-s + (−0.424 + 3.13i)10-s + (0.290 − 1.08i)11-s + (0.548 − 1.49i)12-s − 6.68i·13-s + (2.66 + 2.62i)14-s + (−0.0733 − 1.77i)15-s + (0.715 − 3.93i)16-s + (−0.386 + 1.44i)17-s + ⋯
L(s)  = 1  + (−0.940 + 0.340i)2-s + (0.398 − 0.229i)3-s + (0.767 − 0.640i)4-s + (0.463 − 0.885i)5-s + (−0.295 + 0.351i)6-s + (−0.431 − 0.902i)7-s + (−0.503 + 0.863i)8-s + (−0.394 + 0.683i)9-s + (−0.134 + 0.990i)10-s + (0.0876 − 0.326i)11-s + (0.158 − 0.431i)12-s − 1.85i·13-s + (0.713 + 0.701i)14-s + (−0.0189 − 0.459i)15-s + (0.178 − 0.983i)16-s + (−0.0937 + 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.239 + 0.970i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.551433 - 0.704136i\)
\(L(\frac12)\) \(\approx\) \(0.551433 - 0.704136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.481i)T \)
5 \( 1 + (-1.03 + 1.98i)T \)
7 \( 1 + (1.14 + 2.38i)T \)
good3 \( 1 + (-0.689 + 0.398i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.290 + 1.08i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + 6.68iT - 13T^{2} \)
17 \( 1 + (0.386 - 1.44i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.744 - 2.77i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.000156 + 0.000583i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.47 + 4.47i)T + 29iT^{2} \)
31 \( 1 + (9.18 - 5.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.807 + 1.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.55T + 41T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + (-3.32 + 0.891i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.87 + 5.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.129 + 0.482i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.79 + 1.01i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.77 - 2.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.39iT - 71T^{2} \)
73 \( 1 + (-3.40 + 12.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.43 - 5.44i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.40iT - 83T^{2} \)
89 \( 1 + (-7.59 - 4.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.1 + 11.1i)T + 97iT^{2} \)
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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

−10.45291534320245547198023208039, −9.531274109529938363353229331201, −8.637524148365039020596043037596, −7.938319971923118730927263353140, −7.28206633616917562259187514229, −5.86146688797048190243245617154, −5.33431119104305595927674668486, −3.52965722871751602291951132436, −2.07146688441631335368335856731, −0.63251825817145379074027997895, 2.02136176060741587974835950426, 2.86260849971999583955058157527, 3.95985000497048591663576291971, 5.86371953093801428728544418795, 6.69072582093904375574637144812, 7.40581121713583980656412668567, 8.892464334668979343248921969518, 9.291051461441830382850474440059, 9.746288331269303575326010879989, 11.11148487155736704860868009332

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