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Properties

Label 2-560-7.4-c3-0-25
Degree $2$
Conductor $560$
Sign $0.861 + 0.508i$
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  + (−4.04 + 7.00i)3-s + (2.5 + 4.33i)5-s + (−18.4 + 0.999i)7-s + (−19.1 − 33.2i)9-s + (−19.1 + 33.1i)11-s − 24.4·13-s − 40.4·15-s + (24.5 − 42.5i)17-s + (−8.94 − 15.4i)19-s + (67.7 − 133. i)21-s + (−46.5 − 80.6i)23-s + (−12.5 + 21.6i)25-s + 91.8·27-s + 122.·29-s + (−102. + 177. i)31-s + ⋯
L(s)  = 1  + (−0.777 + 1.34i)3-s + (0.223 + 0.387i)5-s + (−0.998 + 0.0539i)7-s + (−0.710 − 1.23i)9-s + (−0.524 + 0.908i)11-s − 0.522·13-s − 0.695·15-s + (0.350 − 0.606i)17-s + (−0.108 − 0.187i)19-s + (0.704 − 1.38i)21-s + (−0.422 − 0.731i)23-s + (−0.100 + 0.173i)25-s + 0.654·27-s + 0.782·29-s + (−0.595 + 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.861 + 0.508i$
Analytic conductor: \(33.0410\)
Root analytic conductor: \(5.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :3/2),\ 0.861 + 0.508i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3551203191\)
\(L(\frac12)\) \(\approx\) \(0.3551203191\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 - 0.999i)T \)
good3 \( 1 + (4.04 - 7.00i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (19.1 - 33.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 24.4T + 2.19e3T^{2} \)
17 \( 1 + (-24.5 + 42.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (8.94 + 15.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (46.5 + 80.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 122.T + 2.43e4T^{2} \)
31 \( 1 + (102. - 177. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (117. + 203. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 507.T + 6.89e4T^{2} \)
43 \( 1 - 477.T + 7.95e4T^{2} \)
47 \( 1 + (-28.4 - 49.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (29.2 - 50.6i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (0.720 - 1.24i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-415. - 720. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-79.1 + 137. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 202.T + 3.57e5T^{2} \)
73 \( 1 + (-384. + 665. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (306. + 531. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 623.T + 5.71e5T^{2} \)
89 \( 1 + (-800. - 1.38e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.37e3T + 9.12e5T^{2} \)
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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

−10.29172689802416607644932449858, −9.708147637267524286410744314362, −8.922023146272850987569283183262, −7.37509858626280854887854251969, −6.52190995967602018874550880312, −5.47031612785266238024059461542, −4.73818005951827413118940495408, −3.64666125812616219473668938836, −2.51694605294967344989218752148, −0.15219911733168798951985663715, 0.867934400516754521745062260243, 2.18642675796018478361271096669, 3.55344847614233379712887484201, 5.25407418341512805847343427920, 5.97824871077144459756650637811, 6.66967727643279575359591462092, 7.65041916378495340659980973373, 8.439447149522761544124904238854, 9.650688018129915053388350645367, 10.48888834127674158698922937883

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