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Properties

Label 2-560-16.13-c1-0-21
Degree $2$
Conductor $560$
Sign $0.817 - 0.575i$
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  + (−1.40 − 0.198i)2-s + (1.92 + 1.92i)3-s + (1.92 + 0.556i)4-s + (0.707 − 0.707i)5-s + (−2.30 − 3.07i)6-s − i·7-s + (−2.57 − 1.16i)8-s + 4.37i·9-s + (−1.13 + 0.849i)10-s + (2.40 − 2.40i)11-s + (2.62 + 4.75i)12-s + (−0.230 − 0.230i)13-s + (−0.198 + 1.40i)14-s + 2.71·15-s + (3.37 + 2.13i)16-s + 3.07·17-s + ⋯
L(s)  = 1  + (−0.990 − 0.140i)2-s + (1.10 + 1.10i)3-s + (0.960 + 0.278i)4-s + (0.316 − 0.316i)5-s + (−0.942 − 1.25i)6-s − 0.377i·7-s + (−0.911 − 0.410i)8-s + 1.45i·9-s + (−0.357 + 0.268i)10-s + (0.725 − 0.725i)11-s + (0.756 + 1.37i)12-s + (−0.0638 − 0.0638i)13-s + (−0.0531 + 0.374i)14-s + 0.701·15-s + (0.844 + 0.534i)16-s + 0.745·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.817 - 0.575i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.817 - 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45734 + 0.461813i\)
\(L(\frac12)\) \(\approx\) \(1.45734 + 0.461813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.198i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-1.92 - 1.92i)T + 3iT^{2} \)
11 \( 1 + (-2.40 + 2.40i)T - 11iT^{2} \)
13 \( 1 + (0.230 + 0.230i)T + 13iT^{2} \)
17 \( 1 - 3.07T + 17T^{2} \)
19 \( 1 + (-4.98 - 4.98i)T + 19iT^{2} \)
23 \( 1 + 2.85iT - 23T^{2} \)
29 \( 1 + (-3.04 - 3.04i)T + 29iT^{2} \)
31 \( 1 + 5.50T + 31T^{2} \)
37 \( 1 + (5.74 - 5.74i)T - 37iT^{2} \)
41 \( 1 - 2.73iT - 41T^{2} \)
43 \( 1 + (1.65 - 1.65i)T - 43iT^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + (-9.26 + 9.26i)T - 53iT^{2} \)
59 \( 1 + (1.23 - 1.23i)T - 59iT^{2} \)
61 \( 1 + (-0.354 - 0.354i)T + 61iT^{2} \)
67 \( 1 + (10.8 + 10.8i)T + 67iT^{2} \)
71 \( 1 + 1.43iT - 71T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 - 8.10T + 79T^{2} \)
83 \( 1 + (0.0148 + 0.0148i)T + 83iT^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 - 7.61T + 97T^{2} \)
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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

−10.30517786802424117895715475820, −9.957277244357142303358825125912, −9.126541950380301052392692768420, −8.459509268397463262608118809134, −7.75357668797748356329764701136, −6.48772840668977420256722705265, −5.20076747880158045462764327311, −3.70390843723805191898698854342, −3.11083003470295771628541809034, −1.46635559385597275494319302925, 1.36161241684370420573620153794, 2.33974170916594239812823201494, 3.33817705141423786663492584757, 5.46131640401513183867948851572, 6.67867019061526011461877575997, 7.25916785900014265593274281909, 7.918357604775777780644650653824, 9.087850126361478193487751508681, 9.322864601175699955752193298123, 10.41647217565937321995115036874

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