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Properties

Label 2-560-140.27-c2-0-46
Degree $2$
Conductor $560$
Sign $-0.868 + 0.494i$
Analytic cond. $15.2588$
Root an. cond. $3.90626$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.85 − 2.85i)3-s + (2.38 − 4.39i)5-s + (−5.83 − 3.87i)7-s − 7.33i·9-s + 11.7i·11-s + (−14.4 − 14.4i)13-s + (−5.74 − 19.3i)15-s + (−5.99 + 5.99i)17-s − 36.6i·19-s + (−27.7 + 5.60i)21-s + (23.3 + 23.3i)23-s + (−13.6 − 20.9i)25-s + (4.76 + 4.76i)27-s − 20.4i·29-s − 26.3·31-s + ⋯
L(s)  = 1  + (0.952 − 0.952i)3-s + (0.476 − 0.878i)5-s + (−0.833 − 0.552i)7-s − 0.814i·9-s + 1.06i·11-s + (−1.11 − 1.11i)13-s + (−0.382 − 1.29i)15-s + (−0.352 + 0.352i)17-s − 1.93i·19-s + (−1.32 + 0.267i)21-s + (1.01 + 1.01i)23-s + (−0.545 − 0.838i)25-s + (0.176 + 0.176i)27-s − 0.706i·29-s − 0.849·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.494i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.868 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.868 + 0.494i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ -0.868 + 0.494i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.907392874\)
\(L(\frac12)\) \(\approx\) \(1.907392874\)
\(L(2)\) 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 \( 1 + (-2.38 + 4.39i)T \)
7 \( 1 + (5.83 + 3.87i)T \)
good3 \( 1 + (-2.85 + 2.85i)T - 9iT^{2} \)
11 \( 1 - 11.7iT - 121T^{2} \)
13 \( 1 + (14.4 + 14.4i)T + 169iT^{2} \)
17 \( 1 + (5.99 - 5.99i)T - 289iT^{2} \)
19 \( 1 + 36.6iT - 361T^{2} \)
23 \( 1 + (-23.3 - 23.3i)T + 529iT^{2} \)
29 \( 1 + 20.4iT - 841T^{2} \)
31 \( 1 + 26.3T + 961T^{2} \)
37 \( 1 + (23.8 + 23.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 4.49iT - 1.68e3T^{2} \)
43 \( 1 + (-16.6 - 16.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-36.6 - 36.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-61.3 + 61.3i)T - 2.80e3iT^{2} \)
59 \( 1 - 43.1iT - 3.48e3T^{2} \)
61 \( 1 + 87.5iT - 3.72e3T^{2} \)
67 \( 1 + (-18.1 + 18.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 2.44iT - 5.04e3T^{2} \)
73 \( 1 + (-48.9 - 48.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 60.5T + 6.24e3T^{2} \)
83 \( 1 + (26.3 - 26.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 150.T + 7.92e3T^{2} \)
97 \( 1 + (4.43 - 4.43i)T - 9.40e3iT^{2} \)
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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

−9.806163607230096701988858687913, −9.347784214567356138193432863819, −8.421462429222953551949109614533, −7.25548650515144388078325383540, −7.08019740372240017564831815577, −5.52514203604626806775933761754, −4.50368761401052904246821898652, −2.98523083363429444371114151617, −2.06002852907268039322365213287, −0.61199732953877119288620849561, 2.29075430304975000215103271783, 3.12846938021909101104398289940, 3.95620877295527026997934401534, 5.38078941503512135415722952940, 6.40206978827745487256513850542, 7.29348874999538902027102485974, 8.739762017359653093920661693672, 9.098859309928966406520949375827, 10.06076122573375431744171182158, 10.50618528953020354165924905502

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