Properties

Label 2-4560-1.1-c1-0-71
Degree $2$
Conductor $4560$
Sign $-1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 3.64·7-s + 9-s − 5.64·11-s − 5.64·13-s + 15-s − 4·17-s + 19-s + 3.64·21-s + 1.29·23-s + 25-s + 27-s − 6.93·29-s − 6·31-s − 5.64·33-s + 3.64·35-s − 1.64·37-s − 5.64·39-s − 4.35·41-s − 0.354·43-s + 45-s − 9.29·47-s + 6.29·49-s − 4·51-s + 0.708·53-s − 5.64·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.37·7-s + 0.333·9-s − 1.70·11-s − 1.56·13-s + 0.258·15-s − 0.970·17-s + 0.229·19-s + 0.795·21-s + 0.269·23-s + 0.200·25-s + 0.192·27-s − 1.28·29-s − 1.07·31-s − 0.982·33-s + 0.616·35-s − 0.270·37-s − 0.904·39-s − 0.680·41-s − 0.0540·43-s + 0.149·45-s − 1.35·47-s + 0.898·49-s − 0.560·51-s + 0.0973·53-s − 0.761·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 - 3.64T + 7T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
13 \( 1 + 5.64T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 + 6.93T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 1.64T + 37T^{2} \)
41 \( 1 + 4.35T + 41T^{2} \)
43 \( 1 + 0.354T + 43T^{2} \)
47 \( 1 + 9.29T + 47T^{2} \)
53 \( 1 - 0.708T + 53T^{2} \)
59 \( 1 + 0.708T + 59T^{2} \)
61 \( 1 + 0.708T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 1.06T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−7.76177688667222254413923350502, −7.59799929049065973024828667310, −6.70058278207679356763305761096, −5.32764120326536005248220903749, −5.15903143566908425991132953718, −4.38349895758822959219982031548, −3.14208681812751923497089405649, −2.25021389140852050145134855535, −1.80967557397962061215883980847, 0, 1.80967557397962061215883980847, 2.25021389140852050145134855535, 3.14208681812751923497089405649, 4.38349895758822959219982031548, 5.15903143566908425991132953718, 5.32764120326536005248220903749, 6.70058278207679356763305761096, 7.59799929049065973024828667310, 7.76177688667222254413923350502

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