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Properties

Label 2-560-7.2-c3-0-36
Degree $2$
Conductor $560$
Sign $-0.999 + 0.000641i$
Analytic cond. $33.0410$
Root an. cond. $5.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.72 − 6.44i)3-s + (2.5 − 4.33i)5-s + (15.4 + 10.2i)7-s + (−14.2 + 24.6i)9-s + (32.6 + 56.5i)11-s − 56.7·13-s − 37.2·15-s + (−54.4 − 94.2i)17-s + (48.5 − 84.0i)19-s + (8.64 − 137. i)21-s + (63.6 − 110. i)23-s + (−12.5 − 21.6i)25-s + 10.8·27-s − 72.5·29-s + (−61.2 − 106. i)31-s + ⋯
L(s)  = 1  + (−0.716 − 1.24i)3-s + (0.223 − 0.387i)5-s + (0.832 + 0.553i)7-s + (−0.526 + 0.912i)9-s + (0.895 + 1.55i)11-s − 1.21·13-s − 0.640·15-s + (−0.776 − 1.34i)17-s + (0.586 − 1.01i)19-s + (0.0898 − 1.43i)21-s + (0.576 − 0.999i)23-s + (−0.100 − 0.173i)25-s + 0.0773·27-s − 0.464·29-s + (−0.354 − 0.614i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000641i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.000641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.000641i$
Analytic conductor: \(33.0410\)
Root analytic conductor: \(5.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :3/2),\ -0.999 + 0.000641i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8840532834\)
\(L(\frac12)\) \(\approx\) \(0.8840532834\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-15.4 - 10.2i)T \)
good3 \( 1 + (3.72 + 6.44i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-32.6 - 56.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 56.7T + 2.19e3T^{2} \)
17 \( 1 + (54.4 + 94.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-48.5 + 84.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-63.6 + 110. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 72.5T + 2.43e4T^{2} \)
31 \( 1 + (61.2 + 106. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (51.7 - 89.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 46.1T + 6.89e4T^{2} \)
43 \( 1 + 334.T + 7.95e4T^{2} \)
47 \( 1 + (155. - 269. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (302. + 523. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (426. + 738. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-76.8 + 133. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-115. - 200. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.07e3T + 3.57e5T^{2} \)
73 \( 1 + (476. + 825. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-37.0 + 64.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 781.T + 5.71e5T^{2} \)
89 \( 1 + (5.04 - 8.74i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 59.8T + 9.12e5T^{2} \)
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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

−9.689818444173567549159330927722, −9.161823104102867239360401564782, −7.85424649906756774255634216326, −7.06408284657718702456742394924, −6.54330019524602338111852315488, −4.99856174720665312164307189554, −4.80807466383985943872657512182, −2.42515938148959830604470613617, −1.64565050842415559173178619048, −0.29022050967809919668198130686, 1.49230310295847017972237633878, 3.41872092318639359870350075329, 4.15960740636298571299343601304, 5.26636133375153924341691448902, 5.94588651415710123305543771496, 7.12271310616057633793597141741, 8.267869571583599265546447509199, 9.246158958290914406816837020072, 10.11029401008239851571498916727, 10.85316719536127620605277254272

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