Properties

Label 2-560-1.1-c5-0-4
Degree $2$
Conductor $560$
Sign $1$
Analytic cond. $89.8149$
Root an. cond. $9.47707$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.08·3-s + 25·5-s − 49·7-s − 206.·9-s − 407.·11-s − 1.07e3·13-s − 152.·15-s − 1.77e3·17-s − 2.15e3·19-s + 298.·21-s + 4.54e3·23-s + 625·25-s + 2.73e3·27-s + 6.11e3·29-s + 2.85e3·31-s + 2.47e3·33-s − 1.22e3·35-s − 7.03e3·37-s + 6.54e3·39-s − 1.12e4·41-s − 1.06e4·43-s − 5.15e3·45-s + 7.63e3·47-s + 2.40e3·49-s + 1.08e4·51-s + 1.92e4·53-s − 1.01e4·55-s + ⋯
L(s)  = 1  − 0.390·3-s + 0.447·5-s − 0.377·7-s − 0.847·9-s − 1.01·11-s − 1.76·13-s − 0.174·15-s − 1.49·17-s − 1.37·19-s + 0.147·21-s + 1.78·23-s + 0.200·25-s + 0.720·27-s + 1.35·29-s + 0.533·31-s + 0.396·33-s − 0.169·35-s − 0.845·37-s + 0.688·39-s − 1.04·41-s − 0.881·43-s − 0.379·45-s + 0.504·47-s + 0.142·49-s + 0.581·51-s + 0.941·53-s − 0.454·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(89.8149\)
Root analytic conductor: \(9.47707\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6317880004\)
\(L(\frac12)\) \(\approx\) \(0.6317880004\)
\(L(\frac{7}{2})\) 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 \( 1 - 25T \)
7 \( 1 + 49T \)
good3 \( 1 + 6.08T + 243T^{2} \)
11 \( 1 + 407.T + 1.61e5T^{2} \)
13 \( 1 + 1.07e3T + 3.71e5T^{2} \)
17 \( 1 + 1.77e3T + 1.41e6T^{2} \)
19 \( 1 + 2.15e3T + 2.47e6T^{2} \)
23 \( 1 - 4.54e3T + 6.43e6T^{2} \)
29 \( 1 - 6.11e3T + 2.05e7T^{2} \)
31 \( 1 - 2.85e3T + 2.86e7T^{2} \)
37 \( 1 + 7.03e3T + 6.93e7T^{2} \)
41 \( 1 + 1.12e4T + 1.15e8T^{2} \)
43 \( 1 + 1.06e4T + 1.47e8T^{2} \)
47 \( 1 - 7.63e3T + 2.29e8T^{2} \)
53 \( 1 - 1.92e4T + 4.18e8T^{2} \)
59 \( 1 - 9.18e3T + 7.14e8T^{2} \)
61 \( 1 + 3.37e4T + 8.44e8T^{2} \)
67 \( 1 - 2.97e4T + 1.35e9T^{2} \)
71 \( 1 - 2.98e4T + 1.80e9T^{2} \)
73 \( 1 + 4.07e4T + 2.07e9T^{2} \)
79 \( 1 - 3.91e4T + 3.07e9T^{2} \)
83 \( 1 + 3.44e4T + 3.93e9T^{2} \)
89 \( 1 + 7.99e3T + 5.58e9T^{2} \)
97 \( 1 + 5.85e4T + 8.58e9T^{2} \)
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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

−10.20496107465501037061805158592, −9.031996867875866119647689969881, −8.385097838644515013525825399867, −7.03541646170714634991833666033, −6.44845129378547931659830279778, −5.19806168986964551973772084274, −4.70295837532267717215162675917, −2.88408555100993352810611309514, −2.27448055344705129253000620562, −0.36857507883335853337765332098, 0.36857507883335853337765332098, 2.27448055344705129253000620562, 2.88408555100993352810611309514, 4.70295837532267717215162675917, 5.19806168986964551973772084274, 6.44845129378547931659830279778, 7.03541646170714634991833666033, 8.385097838644515013525825399867, 9.031996867875866119647689969881, 10.20496107465501037061805158592

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