Properties

Label 2-560-5.4-c1-0-6
Degree $2$
Conductor $560$
Sign $0.894 - 0.447i$
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 + 2i)5-s i·7-s + 3·9-s − 4i·13-s + 4i·17-s + 4·19-s + 8i·23-s + (−3 + 4i)25-s − 2·29-s + 8·31-s + (2 − i)35-s − 8i·37-s + 6·41-s + 8i·43-s + (3 + 6i)45-s + ⋯
L(s)  = 1  + (0.447 + 0.894i)5-s − 0.377i·7-s + 9-s − 1.10i·13-s + 0.970i·17-s + 0.917·19-s + 1.66i·23-s + (−0.600 + 0.800i)25-s − 0.371·29-s + 1.43·31-s + (0.338 − 0.169i)35-s − 1.31i·37-s + 0.937·41-s + 1.21i·43-s + (0.447 + 0.894i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64805 + 0.389053i\)
\(L(\frac12)\) \(\approx\) \(1.64805 + 0.389053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1 - 2i)T \)
7 \( 1 + iT \)
good3 \( 1 - 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−10.64621296412020896649484374983, −10.06291567971881036302637130340, −9.355483783050034522504474973207, −7.83201703740521485878140838163, −7.38009549533829100373763125701, −6.27437252788839500920666443081, −5.40777674015997082377076974892, −4.00101149824062534662943114451, −3.02442353022434026574576407813, −1.48518091307567065337490613066, 1.21483707228753499924510782515, 2.57548060170950369116985658849, 4.32812241137330901760803721151, 4.90098488757384188230680961111, 6.14924573008747085200813762455, 7.03747731098302302180039020275, 8.135948951116062401115164912635, 9.134949576564553691895301969857, 9.624704648885627346441251697559, 10.55710139719972886206778780483

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