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Properties

Label 2-560-140.139-c3-0-20
Degree $2$
Conductor $560$
Sign $-0.866 - 0.5i$
Analytic cond. $33.0410$
Root an. cond. $5.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.10i·3-s − 11.1·5-s + 18.5i·7-s + 0.987·9-s + 56.4i·11-s + 93.4·13-s − 57.0i·15-s + 133.·17-s − 94.4·21-s + 125.·25-s + 142. i·27-s − 293.·29-s − 287.·33-s − 207. i·35-s + 476. i·39-s + ⋯
L(s)  = 1  + 0.981i·3-s − 0.999·5-s + 0.999i·7-s + 0.0365·9-s + 1.54i·11-s + 1.99·13-s − 0.981i·15-s + 1.91·17-s − 0.981·21-s + 1.00·25-s + 1.01i·27-s − 1.87·29-s − 1.51·33-s − 0.999i·35-s + 1.95i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(33.0410\)
Root analytic conductor: \(5.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :3/2),\ -0.866 - 0.5i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.773106712\)
\(L(\frac12)\) \(\approx\) \(1.773106712\)
\(L(\frac{5}{2})\) 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 \( 1 + 11.1T \)
7 \( 1 - 18.5iT \)
good3 \( 1 - 5.10iT - 27T^{2} \)
11 \( 1 - 56.4iT - 1.33e3T^{2} \)
13 \( 1 - 93.4T + 2.19e3T^{2} \)
17 \( 1 - 133.T + 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 293.T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + 7.95e4T^{2} \)
47 \( 1 - 464. iT - 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 3.00e5T^{2} \)
71 \( 1 + 863. iT - 3.57e5T^{2} \)
73 \( 1 - 523.T + 3.89e5T^{2} \)
79 \( 1 - 487. iT - 4.93e5T^{2} \)
83 \( 1 - 47.6iT - 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 - 8.91T + 9.12e5T^{2} \)
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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

−10.75366304558648414557700993265, −9.754641005206173428733307805646, −9.127434462707278225831011542607, −8.113666150897531841887385373771, −7.31574256708600488001359512072, −5.96599745133750317687164246027, −5.01214383094717871361788175447, −3.98686749920909376786532335788, −3.30174515804012075183585366124, −1.50141352032545687751640243713, 0.64768688030830489541514491687, 1.28426465888549910823820143316, 3.48045641947450694057927250443, 3.78233804435459338053449594835, 5.57014327699056540694881045613, 6.46323392091214495684732360586, 7.46668484375992013001999538963, 8.007097840826949105235724318583, 8.756010834562195697058519201411, 10.21009306719603126530309114071

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