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Properties

Label 2-560-112.53-c1-0-20
Degree $2$
Conductor $560$
Sign $-0.615 - 0.788i$
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  + (1.24 + 0.665i)2-s + (0.412 + 1.53i)3-s + (1.11 + 1.66i)4-s + (−0.258 + 0.965i)5-s + (−0.509 + 2.19i)6-s + (−2.59 + 0.524i)7-s + (0.284 + 2.81i)8-s + (0.399 − 0.230i)9-s + (−0.965 + 1.03i)10-s + (4.57 − 1.22i)11-s + (−2.09 + 2.39i)12-s + (−2.24 + 2.24i)13-s + (−3.58 − 1.07i)14-s − 1.59·15-s + (−1.51 + 3.70i)16-s + (1.27 − 2.20i)17-s + ⋯
L(s)  = 1  + (0.882 + 0.470i)2-s + (0.238 + 0.888i)3-s + (0.556 + 0.830i)4-s + (−0.115 + 0.431i)5-s + (−0.208 + 0.896i)6-s + (−0.980 + 0.198i)7-s + (0.100 + 0.994i)8-s + (0.133 − 0.0768i)9-s + (−0.305 + 0.326i)10-s + (1.37 − 0.369i)11-s + (−0.605 + 0.692i)12-s + (−0.622 + 0.622i)13-s + (−0.958 − 0.286i)14-s − 0.411·15-s + (−0.379 + 0.925i)16-s + (0.308 − 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06347 + 2.18037i\)
\(L(\frac12)\) \(\approx\) \(1.06347 + 2.18037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.665i)T \)
5 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (2.59 - 0.524i)T \)
good3 \( 1 + (-0.412 - 1.53i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-4.57 + 1.22i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.24 - 2.24i)T - 13iT^{2} \)
17 \( 1 + (-1.27 + 2.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.59 + 0.963i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.29 - 1.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.51 + 3.51i)T - 29iT^{2} \)
31 \( 1 + (1.02 - 1.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.295 + 1.10i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 9.11iT - 41T^{2} \)
43 \( 1 + (-3.62 - 3.62i)T + 43iT^{2} \)
47 \( 1 + (1.07 + 1.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.43 + 2.52i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.44 + 1.72i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.92 - 2.65i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.99 + 14.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.0iT - 71T^{2} \)
73 \( 1 + (-5.41 - 3.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.54 - 7.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.5 - 12.5i)T - 83iT^{2} \)
89 \( 1 + (1.45 - 0.841i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−11.22841847794959777195964951593, −10.09483205984094186073031018094, −9.355185053944215737700496102647, −8.521781187698283589386679847507, −7.01360402398614484956184060470, −6.64229110959321703942491653811, −5.51087927759831466233634557709, −4.16663124144251073599926110026, −3.74514124843244130690455958609, −2.54965818779768615187439703802, 1.11476688972243896245198996502, 2.36695854936163203135787659222, 3.68762901561897779414698449532, 4.56977388478788828113078443589, 5.98373999199538751102337312380, 6.67842927792483378883308092129, 7.47541107422006902828497812091, 8.711423527389160237640353878421, 9.850337070291923840430571231593, 10.39458984884777169195533687264

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