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Properties

Label 2-560-560.157-c1-0-15
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: $\chi_{560} (157, \cdot )$
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

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

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