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Properties

Label 2-560-112.109-c1-0-0
Degree $2$
Conductor $560$
Sign $-0.975 + 0.221i$
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  + (1.21 + 0.719i)2-s + (−1.32 + 0.355i)3-s + (0.964 + 1.75i)4-s + (−0.965 − 0.258i)5-s + (−1.87 − 0.521i)6-s + (−1.74 − 1.98i)7-s + (−0.0865 + 2.82i)8-s + (−0.964 + 0.557i)9-s + (−0.989 − 1.01i)10-s + (0.981 + 3.66i)11-s + (−1.90 − 1.98i)12-s + (−4.73 − 4.73i)13-s + (−0.692 − 3.67i)14-s + 1.37·15-s + (−2.13 + 3.37i)16-s + (−3.90 + 6.77i)17-s + ⋯
L(s)  = 1  + (0.860 + 0.508i)2-s + (−0.765 + 0.205i)3-s + (0.482 + 0.876i)4-s + (−0.431 − 0.115i)5-s + (−0.763 − 0.213i)6-s + (−0.659 − 0.751i)7-s + (−0.0306 + 0.999i)8-s + (−0.321 + 0.185i)9-s + (−0.312 − 0.319i)10-s + (0.296 + 1.10i)11-s + (−0.549 − 0.571i)12-s + (−1.31 − 1.31i)13-s + (−0.185 − 0.982i)14-s + 0.354·15-s + (−0.534 + 0.844i)16-s + (−0.948 + 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.975 + 0.221i$
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: \((2,\ 560,\ (\ :1/2),\ -0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0642195 - 0.573274i\)
\(L(\frac12)\) \(\approx\) \(0.0642195 - 0.573274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 - 0.719i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (1.74 + 1.98i)T \)
good3 \( 1 + (1.32 - 0.355i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.981 - 3.66i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.73 + 4.73i)T + 13iT^{2} \)
17 \( 1 + (3.90 - 6.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0865 - 0.323i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.15 - 1.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.49 + 3.49i)T + 29iT^{2} \)
31 \( 1 + (-4.24 + 7.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.430 + 0.115i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.90iT - 41T^{2} \)
43 \( 1 + (0.00851 - 0.00851i)T - 43iT^{2} \)
47 \( 1 + (-3.97 - 6.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.546 + 2.04i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.243 + 0.908i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.83 - 10.5i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-9.67 + 2.59i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.10iT - 71T^{2} \)
73 \( 1 + (6.14 + 3.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.31 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.79 + 2.79i)T + 83iT^{2} \)
89 \( 1 + (-10.4 + 6.01i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.17T + 97T^{2} \)
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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

−11.35766715500688895000661283139, −10.51340298349726311647202964610, −9.712894900607504046415718380307, −8.139382472433314874761069301940, −7.54824381228152025874809404727, −6.50857848113915286150669803096, −5.74756203911057767226944634569, −4.62453029756987433213181289870, −3.99827195857946686382173476695, −2.54121323621817151341371953813, 0.25044537193009463745730412060, 2.38224752000770877507201228770, 3.38498459263122325570253976731, 4.74101828955596421830948313861, 5.51356514635506147237224974186, 6.61596007080523328826869562357, 6.96785684277986878942733594094, 8.870019185948953648376999174182, 9.433952876347810166740746723383, 10.70309502866704208421785991454

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