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Properties

Label 2-560-28.3-c1-0-11
Degree $2$
Conductor $560$
Sign $0.686 + 0.727i$
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  + (−1.60 + 2.78i)3-s + (0.866 − 0.5i)5-s + (−0.279 − 2.63i)7-s + (−3.68 − 6.37i)9-s + (1.40 + 0.808i)11-s − 5.32i·13-s + 3.21i·15-s + (−4.03 − 2.33i)17-s + (−2.68 − 4.64i)19-s + (7.78 + 3.45i)21-s + (−5.13 + 2.96i)23-s + (0.499 − 0.866i)25-s + 14.0·27-s + 8.25·29-s + (2.66 − 4.61i)31-s + ⋯
L(s)  = 1  + (−0.929 + 1.60i)3-s + (0.387 − 0.223i)5-s + (−0.105 − 0.994i)7-s + (−1.22 − 2.12i)9-s + (0.422 + 0.243i)11-s − 1.47i·13-s + 0.831i·15-s + (−0.978 − 0.565i)17-s + (−0.615 − 1.06i)19-s + (1.69 + 0.754i)21-s + (−1.07 + 0.617i)23-s + (0.0999 − 0.173i)25-s + 2.70·27-s + 1.53·29-s + (0.478 − 0.829i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706424 - 0.304756i\)
\(L(\frac12)\) \(\approx\) \(0.706424 - 0.304756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.279 + 2.63i)T \)
good3 \( 1 + (1.60 - 2.78i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.40 - 0.808i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.32iT - 13T^{2} \)
17 \( 1 + (4.03 + 2.33i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.68 + 4.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.13 - 2.96i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.25T + 29T^{2} \)
31 \( 1 + (-2.66 + 4.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.02 - 5.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.35iT - 41T^{2} \)
43 \( 1 - 3.04iT - 43T^{2} \)
47 \( 1 + (1.08 + 1.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.69 - 4.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.72 + 6.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.549 - 0.317i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.3 + 7.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.61iT - 71T^{2} \)
73 \( 1 + (-0.645 - 0.372i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.535 - 0.309i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 + (1.96 - 1.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.11iT - 97T^{2} \)
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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

−10.54532415842549222890241174020, −9.948558357125092891762179005133, −9.291611788900084301084751455354, −8.162091862047151238299772220075, −6.70953180844589592761161774407, −5.92504900200778031460809553368, −4.78106973717461520208871069460, −4.31823950427103511525597156320, −3.05431840577983199959347028207, −0.49245910845772014012825548419, 1.63682461530733884610398423494, 2.41273680931775444272825133660, 4.46194346457346291973464988585, 5.85514837108622630694577546046, 6.33170677271497401241688967022, 6.89990708935298366815382954615, 8.241311719627781573553951398518, 8.814539653342923445810035444741, 10.19845538958748024511568234476, 11.17209915541364366516144709848

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