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Properties

Label 2-560-112.109-c1-0-37
Degree $2$
Conductor $560$
Sign $0.836 - 0.547i$
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  + (−1.11 + 0.872i)2-s + (3.17 − 0.849i)3-s + (0.476 − 1.94i)4-s + (0.965 + 0.258i)5-s + (−2.78 + 3.71i)6-s + (−1.44 + 2.21i)7-s + (1.16 + 2.57i)8-s + (6.73 − 3.89i)9-s + (−1.30 + 0.555i)10-s + (1.12 + 4.18i)11-s + (−0.139 − 6.56i)12-s + (−0.203 − 0.203i)13-s + (−0.326 − 3.72i)14-s + 3.28·15-s + (−3.54 − 1.85i)16-s + (2.16 − 3.75i)17-s + ⋯
L(s)  = 1  + (−0.786 + 0.617i)2-s + (1.83 − 0.490i)3-s + (0.238 − 0.971i)4-s + (0.431 + 0.115i)5-s + (−1.13 + 1.51i)6-s + (−0.546 + 0.837i)7-s + (0.412 + 0.911i)8-s + (2.24 − 1.29i)9-s + (−0.411 + 0.175i)10-s + (0.337 + 1.26i)11-s + (−0.0404 − 1.89i)12-s + (−0.0563 − 0.0563i)13-s + (−0.0871 − 0.996i)14-s + 0.847·15-s + (−0.886 − 0.462i)16-s + (0.525 − 0.910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.836 - 0.547i$
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: \((2,\ 560,\ (\ :1/2),\ 0.836 - 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84864 + 0.551127i\)
\(L(\frac12)\) \(\approx\) \(1.84864 + 0.551127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.872i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (1.44 - 2.21i)T \)
good3 \( 1 + (-3.17 + 0.849i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.12 - 4.18i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.203 + 0.203i)T + 13iT^{2} \)
17 \( 1 + (-2.16 + 3.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.82 - 6.79i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.32 - 0.767i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.13 + 6.13i)T + 29iT^{2} \)
31 \( 1 + (-2.31 + 4.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.41 + 0.379i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.24iT - 41T^{2} \)
43 \( 1 + (-6.01 + 6.01i)T - 43iT^{2} \)
47 \( 1 + (5.12 + 8.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.108 + 0.405i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.30 + 4.86i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.05 - 7.67i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.742 + 0.198i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.54iT - 71T^{2} \)
73 \( 1 + (1.53 + 0.886i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.58 - 9.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.02 + 4.02i)T + 83iT^{2} \)
89 \( 1 + (10.5 - 6.09i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.806T + 97T^{2} \)
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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

−10.00164432726124835117293579555, −9.767740339009918180352374301105, −9.078671181356551117236570132457, −8.188014546972404608717118135419, −7.47940357762800048611198466941, −6.70474255893146943696520192083, −5.62966473364465258048338701224, −3.95198911539113037214338456371, −2.47941286539387640017430813537, −1.79676422390029558423169069243, 1.43080162352038931123936537343, 2.86704859691431462955619886753, 3.47592837290354602784878154342, 4.45668655308371423466965939954, 6.52510933693206721783142802862, 7.53226297824706823525660826135, 8.350434881196026400797941584883, 9.077379916244821695962902749292, 9.549499343520886429783576838324, 10.53034348105090231268977047238

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