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Properties

Label 2-560-112.109-c1-0-12
Degree $2$
Conductor $560$
Sign $0.990 + 0.138i$
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  + (−0.246 − 1.39i)2-s + (−1.41 + 0.379i)3-s + (−1.87 + 0.685i)4-s + (−0.965 − 0.258i)5-s + (0.876 + 1.87i)6-s + (−1.63 − 2.08i)7-s + (1.41 + 2.44i)8-s + (−0.739 + 0.426i)9-s + (−0.122 + 1.40i)10-s + (−0.00275 − 0.0102i)11-s + (2.39 − 1.68i)12-s + (1.75 + 1.75i)13-s + (−2.49 + 2.78i)14-s + 1.46·15-s + (3.06 − 2.57i)16-s + (−0.216 + 0.375i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.816 + 0.218i)3-s + (−0.939 + 0.342i)4-s + (−0.431 − 0.115i)5-s + (0.357 + 0.766i)6-s + (−0.617 − 0.786i)7-s + (0.500 + 0.865i)8-s + (−0.246 + 0.142i)9-s + (−0.0388 + 0.445i)10-s + (−0.000830 − 0.00309i)11-s + (0.692 − 0.485i)12-s + (0.485 + 0.485i)13-s + (−0.667 + 0.744i)14-s + 0.378·15-s + (0.765 − 0.643i)16-s + (−0.0525 + 0.0910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.990 + 0.138i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607087 - 0.0421340i\)
\(L(\frac12)\) \(\approx\) \(0.607087 - 0.0421340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.246 + 1.39i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (1.63 + 2.08i)T \)
good3 \( 1 + (1.41 - 0.379i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.00275 + 0.0102i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.75 - 1.75i)T + 13iT^{2} \)
17 \( 1 + (0.216 - 0.375i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.646 - 2.41i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.50 + 2.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.06 - 6.06i)T + 29iT^{2} \)
31 \( 1 + (1.48 - 2.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.70 - 1.79i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.88iT - 41T^{2} \)
43 \( 1 + (-6.27 + 6.27i)T - 43iT^{2} \)
47 \( 1 + (-1.14 - 1.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.875 - 3.26i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.05 + 3.92i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.50 - 5.62i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.29 - 1.42i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + (-1.20 - 0.696i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.04 + 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.86 - 1.86i)T + 83iT^{2} \)
89 \( 1 + (7.02 - 4.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.20T + 97T^{2} \)
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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

−10.72467089255813341447773025404, −10.28259988096457214665620597595, −9.131372885904962265519042025719, −8.342674133391369953323548846569, −7.17728426970788720999561541372, −6.04985192718421759729211202731, −4.83901388774187317176124492843, −4.03358152828869995422876580314, −2.90916600457305495132257175470, −0.999395237683299151215236454670, 0.55545827563958442019891035734, 3.04087552987909278289261801266, 4.47139763947661653758119710185, 5.61961787201173219528923722914, 6.16013445840254072836043143186, 6.99656471585861108710691241627, 8.054338874824933307910990871984, 8.907145782814057352925837651031, 9.690208557466894383745551091409, 10.83864884929119304929882567218

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