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Properties

Label 2-560-140.27-c2-0-11
Degree $2$
Conductor $560$
Sign $0.219 - 0.975i$
Analytic cond. $15.2588$
Root an. cond. $3.90626$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.904 + 0.904i)3-s + (−2.12 − 4.52i)5-s + (5.39 + 4.46i)7-s + 7.36i·9-s − 1.26i·11-s + (−5.68 − 5.68i)13-s + (6.01 + 2.16i)15-s + (−3.36 + 3.36i)17-s − 10.5i·19-s + (−8.91 + 0.844i)21-s + (17.3 + 17.3i)23-s + (−15.9 + 19.2i)25-s + (−14.7 − 14.7i)27-s + 33.6i·29-s + 19.7·31-s + ⋯
L(s)  = 1  + (−0.301 + 0.301i)3-s + (−0.425 − 0.905i)5-s + (0.770 + 0.637i)7-s + 0.818i·9-s − 0.115i·11-s + (−0.437 − 0.437i)13-s + (0.401 + 0.144i)15-s + (−0.198 + 0.198i)17-s − 0.555i·19-s + (−0.424 + 0.0402i)21-s + (0.755 + 0.755i)23-s + (−0.638 + 0.769i)25-s + (−0.548 − 0.548i)27-s + 1.16i·29-s + 0.636·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.219 - 0.975i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ 0.219 - 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.292352117\)
\(L(\frac12)\) \(\approx\) \(1.292352117\)
\(L(2)\) 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 \( 1 + (2.12 + 4.52i)T \)
7 \( 1 + (-5.39 - 4.46i)T \)
good3 \( 1 + (0.904 - 0.904i)T - 9iT^{2} \)
11 \( 1 + 1.26iT - 121T^{2} \)
13 \( 1 + (5.68 + 5.68i)T + 169iT^{2} \)
17 \( 1 + (3.36 - 3.36i)T - 289iT^{2} \)
19 \( 1 + 10.5iT - 361T^{2} \)
23 \( 1 + (-17.3 - 17.3i)T + 529iT^{2} \)
29 \( 1 - 33.6iT - 841T^{2} \)
31 \( 1 - 19.7T + 961T^{2} \)
37 \( 1 + (-23.2 - 23.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 44.1iT - 1.68e3T^{2} \)
43 \( 1 + (-34.3 - 34.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-15.2 - 15.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (44.2 - 44.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 46.7iT - 3.48e3T^{2} \)
61 \( 1 + 12.0iT - 3.72e3T^{2} \)
67 \( 1 + (-49.0 + 49.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 98.6iT - 5.04e3T^{2} \)
73 \( 1 + (48.8 + 48.8i)T + 5.32e3iT^{2} \)
79 \( 1 - 13.3T + 6.24e3T^{2} \)
83 \( 1 + (41.6 - 41.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 110.T + 7.92e3T^{2} \)
97 \( 1 + (-54.3 + 54.3i)T - 9.40e3iT^{2} \)
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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

−11.03586364376880740050763329980, −9.812491582677374998432890686910, −8.907192379529281179838509461377, −8.138302960880963432496642772224, −7.41129715813919971871628146810, −5.86220829581705203139823781946, −4.99144513833693809663624580842, −4.50211758488708551807266960273, −2.82005093400577876326442504950, −1.31543020660184278444212595725, 0.56775788547087660947190526881, 2.24546817265985573617952854870, 3.67266554093886912269567396244, 4.55774623741655001404072492564, 5.95411183183376733914379549660, 6.89744523938971254092002739362, 7.45068798117444595713875119100, 8.464972376896291739074385521718, 9.633357642529676745910055223504, 10.51055560482539494377803862897

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