Properties

Label 2-560-5.4-c3-0-48
Degree $2$
Conductor $560$
Sign $-0.978 + 0.208i$
Analytic cond. $33.0410$
Root an. cond. $5.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93i·3-s + (−2.32 − 10.9i)5-s + 7i·7-s + 23.2·9-s − 25.5·11-s − 64.1i·13-s + (−21.2 + 4.51i)15-s + 27.6i·17-s + 0.792·19-s + 13.5·21-s − 108. i·23-s + (−114. + 50.9i)25-s − 97.4i·27-s + 234.·29-s − 129.·31-s + ⋯
L(s)  = 1  − 0.373i·3-s + (−0.208 − 0.978i)5-s + 0.377i·7-s + 0.860·9-s − 0.700·11-s − 1.36i·13-s + (−0.365 + 0.0777i)15-s + 0.395i·17-s + 0.00956·19-s + 0.141·21-s − 0.984i·23-s + (−0.913 + 0.407i)25-s − 0.694i·27-s + 1.49·29-s − 0.748·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.978 + 0.208i$
Analytic conductor: \(33.0410\)
Root analytic conductor: \(5.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :3/2),\ -0.978 + 0.208i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.053696768\)
\(L(\frac12)\) \(\approx\) \(1.053696768\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.32 + 10.9i)T \)
7 \( 1 - 7iT \)
good3 \( 1 + 1.93iT - 27T^{2} \)
11 \( 1 + 25.5T + 1.33e3T^{2} \)
13 \( 1 + 64.1iT - 2.19e3T^{2} \)
17 \( 1 - 27.6iT - 4.91e3T^{2} \)
19 \( 1 - 0.792T + 6.85e3T^{2} \)
23 \( 1 + 108. iT - 1.21e4T^{2} \)
29 \( 1 - 234.T + 2.43e4T^{2} \)
31 \( 1 + 129.T + 2.97e4T^{2} \)
37 \( 1 + 38.3iT - 5.06e4T^{2} \)
41 \( 1 + 403.T + 6.89e4T^{2} \)
43 \( 1 + 172. iT - 7.95e4T^{2} \)
47 \( 1 - 206. iT - 1.03e5T^{2} \)
53 \( 1 - 144. iT - 1.48e5T^{2} \)
59 \( 1 + 679.T + 2.05e5T^{2} \)
61 \( 1 + 574.T + 2.26e5T^{2} \)
67 \( 1 - 515. iT - 3.00e5T^{2} \)
71 \( 1 + 556.T + 3.57e5T^{2} \)
73 \( 1 + 173. iT - 3.89e5T^{2} \)
79 \( 1 - 79.3T + 4.93e5T^{2} \)
83 \( 1 + 1.04e3iT - 5.71e5T^{2} \)
89 \( 1 + 652.T + 7.04e5T^{2} \)
97 \( 1 + 515. iT - 9.12e5T^{2} \)
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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

−10.08178153434124443154038945489, −8.876405934548391451822673360952, −8.163002289574597322180443585377, −7.45228763522908269927513873875, −6.20734001431213556595844731131, −5.22335469470988763284171382873, −4.38294510986031494933877223390, −2.95517687442626099006353183729, −1.54355248770685609753885560826, −0.31320545496437749457231338719, 1.68927878622023975253544524171, 3.08580647142355475599630090587, 4.09575015570170317116377936919, 5.02038843069162909412396264012, 6.47603192646763346997200925931, 7.11222044747110897582088568461, 7.929597203833289142400525614325, 9.232482543492643403805239780460, 10.00774034313717126805884330753, 10.66603300570194355340319325960

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