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Properties

Label 2-560-5.4-c1-0-0
Degree $2$
Conductor $560$
Sign $-0.621 - 0.783i$
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  + 0.363i·3-s + (−1.75 + 1.38i)5-s − i·7-s + 2.86·9-s − 5.14·11-s + 4.64i·13-s + (−0.504 − 0.636i)15-s + 3.86i·17-s − 0.778·19-s + 0.363·21-s + 5.00i·23-s + (1.14 − 4.86i)25-s + 2.13i·27-s − 9.42·29-s − 4.72·31-s + ⋯
L(s)  = 1  + 0.209i·3-s + (−0.783 + 0.621i)5-s − 0.377i·7-s + 0.955·9-s − 1.55·11-s + 1.28i·13-s + (−0.130 − 0.164i)15-s + 0.938i·17-s − 0.178·19-s + 0.0792·21-s + 1.04i·23-s + (0.228 − 0.973i)25-s + 0.410i·27-s − 1.74·29-s − 0.848·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.621 - 0.783i$
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: \((2,\ 560,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349946 + 0.723937i\)
\(L(\frac12)\) \(\approx\) \(0.349946 + 0.723937i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.75 - 1.38i)T \)
7 \( 1 + iT \)
good3 \( 1 - 0.363iT - 3T^{2} \)
11 \( 1 + 5.14T + 11T^{2} \)
13 \( 1 - 4.64iT - 13T^{2} \)
17 \( 1 - 3.86iT - 17T^{2} \)
19 \( 1 + 0.778T + 19T^{2} \)
23 \( 1 - 5.00iT - 23T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + 4.72T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 1.00T + 41T^{2} \)
43 \( 1 + 7.00iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 2.72T + 71T^{2} \)
73 \( 1 + 5.00iT - 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 + 4.67iT - 83T^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 + 1.58iT - 97T^{2} \)
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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

−10.98244029861962307417181766209, −10.32441230253612989618182669878, −9.490734512353544634730230054291, −8.248203073897564330384074295322, −7.42974978681483996921058270604, −6.84027301768124168392837837516, −5.46359701149597170961347255797, −4.26698219879601631519074926766, −3.55264762165505726585407964457, −1.96853018671105723091098124064, 0.44648047525435250959307692164, 2.35820578998841619374724966566, 3.68748847284982373605930481326, 4.94506077533364295543517149083, 5.56398925708776668727302718891, 7.16201970589236495679756126986, 7.74384845516185904536577837324, 8.512168878131937771699206445227, 9.589074022659839418017664758910, 10.51122538332543913015797817205

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