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Properties

Label 2-560-112.109-c1-0-22
Degree $2$
Conductor $560$
Sign $0.983 + 0.178i$
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  + (−1.41 − 0.0823i)2-s + (−2.87 + 0.770i)3-s + (1.98 + 0.232i)4-s + (0.965 + 0.258i)5-s + (4.12 − 0.850i)6-s + (2.64 + 0.0382i)7-s + (−2.78 − 0.491i)8-s + (5.06 − 2.92i)9-s + (−1.34 − 0.444i)10-s + (0.612 + 2.28i)11-s + (−5.88 + 0.861i)12-s + (−4.47 − 4.47i)13-s + (−3.73 − 0.271i)14-s − 2.97·15-s + (3.89 + 0.923i)16-s + (2.10 − 3.64i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0582i)2-s + (−1.65 + 0.444i)3-s + (0.993 + 0.116i)4-s + (0.431 + 0.115i)5-s + (1.68 − 0.347i)6-s + (0.999 + 0.0144i)7-s + (−0.984 − 0.173i)8-s + (1.68 − 0.975i)9-s + (−0.424 − 0.140i)10-s + (0.184 + 0.689i)11-s + (−1.69 + 0.248i)12-s + (−1.24 − 1.24i)13-s + (−0.997 − 0.0726i)14-s − 0.768·15-s + (0.972 + 0.230i)16-s + (0.510 − 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.983 + 0.178i$
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: \((2,\ 560,\ (\ :1/2),\ 0.983 + 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592422 - 0.0534336i\)
\(L(\frac12)\) \(\approx\) \(0.592422 - 0.0534336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0823i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-2.64 - 0.0382i)T \)
good3 \( 1 + (2.87 - 0.770i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.612 - 2.28i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.47 + 4.47i)T + 13iT^{2} \)
17 \( 1 + (-2.10 + 3.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.669 - 2.49i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.85 - 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.56 + 2.56i)T + 29iT^{2} \)
31 \( 1 + (-4.00 + 6.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.26 - 2.48i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.09iT - 41T^{2} \)
43 \( 1 + (-7.60 + 7.60i)T - 43iT^{2} \)
47 \( 1 + (-4.43 - 7.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.82 - 10.5i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.752 + 2.80i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.22 + 8.30i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.0561 - 0.0150i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.37iT - 71T^{2} \)
73 \( 1 + (-3.33 - 1.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 + 2.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.720 + 0.720i)T + 83iT^{2} \)
89 \( 1 + (-4.71 + 2.72i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.66T + 97T^{2} \)
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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

−10.61697570077568338603660815497, −9.985260957524420263360466443667, −9.449516240023899121654795716424, −7.80646998431744704584900104198, −7.34544012909327968937525182017, −5.99942786119792183008732684094, −5.46043309494081903573152738372, −4.36947994646565903246404333709, −2.34821116873045343064291222475, −0.75392425248951671584875428473, 1.03059112622666119232546671326, 2.13780617530931285670246204519, 4.53871267545409438544046430995, 5.52855197870045013825632706423, 6.33896348514210933912704333035, 7.09765795638550395913495705265, 8.029383178989854641189184953440, 9.095937429163879983843129719288, 10.13199067207854969126673433378, 10.82657307379766603743222963858

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