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Properties

Label 2-560-112.109-c1-0-4
Degree $2$
Conductor $560$
Sign $0.142 - 0.989i$
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  + (0.909 − 1.08i)2-s + (−1.83 + 0.492i)3-s + (−0.345 − 1.96i)4-s + (0.965 + 0.258i)5-s + (−1.13 + 2.43i)6-s + (−1.40 + 2.24i)7-s + (−2.44 − 1.41i)8-s + (0.534 − 0.308i)9-s + (1.15 − 0.810i)10-s + (1.01 + 3.77i)11-s + (1.60 + 3.44i)12-s + (−3.90 − 3.90i)13-s + (1.15 + 3.55i)14-s − 1.90·15-s + (−3.76 + 1.36i)16-s + (−1.98 + 3.43i)17-s + ⋯
L(s)  = 1  + (0.643 − 0.765i)2-s + (−1.06 + 0.284i)3-s + (−0.172 − 0.984i)4-s + (0.431 + 0.115i)5-s + (−0.464 + 0.995i)6-s + (−0.529 + 0.848i)7-s + (−0.865 − 0.500i)8-s + (0.178 − 0.102i)9-s + (0.366 − 0.256i)10-s + (0.305 + 1.13i)11-s + (0.463 + 0.995i)12-s + (−1.08 − 1.08i)13-s + (0.308 + 0.951i)14-s − 0.491·15-s + (−0.940 + 0.340i)16-s + (−0.480 + 0.832i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.142 - 0.989i$
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: \((2,\ 560,\ (\ :1/2),\ 0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516911 + 0.447749i\)
\(L(\frac12)\) \(\approx\) \(0.516911 + 0.447749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 + 1.08i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (1.40 - 2.24i)T \)
good3 \( 1 + (1.83 - 0.492i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.01 - 3.77i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.90 + 3.90i)T + 13iT^{2} \)
17 \( 1 + (1.98 - 3.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.86 - 6.96i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.79 + 2.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.00116 - 0.00116i)T + 29iT^{2} \)
31 \( 1 + (5.27 - 9.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.81 + 0.753i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.98iT - 41T^{2} \)
43 \( 1 + (-3.18 + 3.18i)T - 43iT^{2} \)
47 \( 1 + (-1.44 - 2.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.392 + 1.46i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.14 + 4.26i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.04 + 3.91i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (9.46 - 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.284iT - 71T^{2} \)
73 \( 1 + (-5.69 - 3.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.47 + 7.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (7.10 - 4.09i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.09T + 97T^{2} \)
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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

−10.85688014853623181178503547930, −10.31375595378149509488002421416, −9.692172379585449572352598905978, −8.603887836862113914715654695611, −6.92617750406158948242651567672, −6.01229753947250255733774760088, −5.36552281308345333423760994207, −4.55349686642710640621898051925, −3.10082906544349713596618687711, −1.89837094684869779018544664026, 0.34147484315641722375768771902, 2.78696739550804881604202410619, 4.21075029494496695491691510090, 5.13487076655910415076380255934, 6.01376300626388298019165756317, 6.91740826504983503552830191448, 7.22018480327376454725548035442, 8.912036762553034966894725353504, 9.422330526865371820886499157533, 11.03732627825623595932578633031

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