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Properties

Label 2-560-16.5-c1-0-31
Degree $2$
Conductor $560$
Sign $-0.855 + 0.517i$
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  + (−0.590 − 1.28i)2-s + (−0.839 + 0.839i)3-s + (−1.30 + 1.51i)4-s + (−0.707 − 0.707i)5-s + (1.57 + 0.583i)6-s + i·7-s + (2.71 + 0.778i)8-s + 1.59i·9-s + (−0.491 + 1.32i)10-s + (−2.36 − 2.36i)11-s + (−0.179 − 2.36i)12-s + (0.604 − 0.604i)13-s + (1.28 − 0.590i)14-s + 1.18·15-s + (−0.604 − 3.95i)16-s − 3.62·17-s + ⋯
L(s)  = 1  + (−0.417 − 0.908i)2-s + (−0.484 + 0.484i)3-s + (−0.651 + 0.758i)4-s + (−0.316 − 0.316i)5-s + (0.642 + 0.238i)6-s + 0.377i·7-s + (0.961 + 0.275i)8-s + 0.530i·9-s + (−0.155 + 0.419i)10-s + (−0.711 − 0.711i)11-s + (−0.0519 − 0.683i)12-s + (0.167 − 0.167i)13-s + (0.343 − 0.157i)14-s + 0.306·15-s + (−0.151 − 0.988i)16-s − 0.878·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.855 + 0.517i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113174 - 0.405509i\)
\(L(\frac12)\) \(\approx\) \(0.113174 - 0.405509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.590 + 1.28i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (0.839 - 0.839i)T - 3iT^{2} \)
11 \( 1 + (2.36 + 2.36i)T + 11iT^{2} \)
13 \( 1 + (-0.604 + 0.604i)T - 13iT^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
19 \( 1 + (-5.04 + 5.04i)T - 19iT^{2} \)
23 \( 1 + 7.53iT - 23T^{2} \)
29 \( 1 + (4.76 - 4.76i)T - 29iT^{2} \)
31 \( 1 - 3.76T + 31T^{2} \)
37 \( 1 + (0.829 + 0.829i)T + 37iT^{2} \)
41 \( 1 + 5.50iT - 41T^{2} \)
43 \( 1 + (7.86 + 7.86i)T + 43iT^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + (8.17 + 8.17i)T + 53iT^{2} \)
59 \( 1 + (-4.06 - 4.06i)T + 59iT^{2} \)
61 \( 1 + (-6.88 + 6.88i)T - 61iT^{2} \)
67 \( 1 + (-9.89 + 9.89i)T - 67iT^{2} \)
71 \( 1 - 5.93iT - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 + 5.60T + 79T^{2} \)
83 \( 1 + (-7.06 + 7.06i)T - 83iT^{2} \)
89 \( 1 + 5.86iT - 89T^{2} \)
97 \( 1 - 7.10T + 97T^{2} \)
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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

−10.59739074878134589134148794723, −9.686961557846292617848234460223, −8.674443477436646272478697579905, −8.160137584881906916740689508561, −6.90840702199786897096738752664, −5.27950272513050303054089832749, −4.79437674435780099659985070409, −3.45664293489704997782770149666, −2.28487588403099071825173407261, −0.30632420663427852795526812088, 1.47314124384549762796698386553, 3.60270273255067234300556025669, 4.84807161689377437184118981634, 5.90187288310452640314340797059, 6.70080433085628003788245317789, 7.53042638599867430163738540632, 8.081273386061720454381919699265, 9.530445060875147711955844826537, 9.920125649398337443702000090498, 11.16989570535609477872922516016

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