Properties

Label 2-504-1.1-c5-0-14
Degree $2$
Conductor $504$
Sign $1$
Analytic cond. $80.8334$
Root an. cond. $8.99074$
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  + 97.5·5-s + 49·7-s − 406.·11-s − 905.·13-s − 359.·17-s + 1.81e3·19-s + 2.16e3·23-s + 6.38e3·25-s + 4.09e3·29-s + 4.45e3·31-s + 4.77e3·35-s + 8.93e3·37-s + 3.41e3·41-s − 8.69e3·43-s − 1.42e4·47-s + 2.40e3·49-s + 1.46e4·53-s − 3.96e4·55-s − 2.13e4·59-s + 5.40e4·61-s − 8.83e4·65-s + 4.49e4·67-s − 5.70e3·71-s + 4.76e4·73-s − 1.99e4·77-s + 5.80e3·79-s − 2.70e4·83-s + ⋯
L(s)  = 1  + 1.74·5-s + 0.377·7-s − 1.01·11-s − 1.48·13-s − 0.301·17-s + 1.15·19-s + 0.852·23-s + 2.04·25-s + 0.903·29-s + 0.831·31-s + 0.659·35-s + 1.07·37-s + 0.317·41-s − 0.717·43-s − 0.937·47-s + 0.142·49-s + 0.714·53-s − 1.76·55-s − 0.800·59-s + 1.86·61-s − 2.59·65-s + 1.22·67-s − 0.134·71-s + 1.04·73-s − 0.382·77-s + 0.104·79-s − 0.431·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(80.8334\)
Root analytic conductor: \(8.99074\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.079509458\)
\(L(\frac12)\) \(\approx\) \(3.079509458\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 49T \)
good5 \( 1 - 97.5T + 3.12e3T^{2} \)
11 \( 1 + 406.T + 1.61e5T^{2} \)
13 \( 1 + 905.T + 3.71e5T^{2} \)
17 \( 1 + 359.T + 1.41e6T^{2} \)
19 \( 1 - 1.81e3T + 2.47e6T^{2} \)
23 \( 1 - 2.16e3T + 6.43e6T^{2} \)
29 \( 1 - 4.09e3T + 2.05e7T^{2} \)
31 \( 1 - 4.45e3T + 2.86e7T^{2} \)
37 \( 1 - 8.93e3T + 6.93e7T^{2} \)
41 \( 1 - 3.41e3T + 1.15e8T^{2} \)
43 \( 1 + 8.69e3T + 1.47e8T^{2} \)
47 \( 1 + 1.42e4T + 2.29e8T^{2} \)
53 \( 1 - 1.46e4T + 4.18e8T^{2} \)
59 \( 1 + 2.13e4T + 7.14e8T^{2} \)
61 \( 1 - 5.40e4T + 8.44e8T^{2} \)
67 \( 1 - 4.49e4T + 1.35e9T^{2} \)
71 \( 1 + 5.70e3T + 1.80e9T^{2} \)
73 \( 1 - 4.76e4T + 2.07e9T^{2} \)
79 \( 1 - 5.80e3T + 3.07e9T^{2} \)
83 \( 1 + 2.70e4T + 3.93e9T^{2} \)
89 \( 1 - 9.62e4T + 5.58e9T^{2} \)
97 \( 1 - 1.01e5T + 8.58e9T^{2} \)
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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

−9.898991370242585629651535132176, −9.591656256696332292364270270137, −8.388355494805211027420834492546, −7.33852326907518346972394024383, −6.38970651874797197063000701809, −5.24261944571308483278222942633, −4.90453838959611915338473952714, −2.84273132491704866433304713472, −2.20794676708162849652633382074, −0.897039847065639093826117081168, 0.897039847065639093826117081168, 2.20794676708162849652633382074, 2.84273132491704866433304713472, 4.90453838959611915338473952714, 5.24261944571308483278222942633, 6.38970651874797197063000701809, 7.33852326907518346972394024383, 8.388355494805211027420834492546, 9.591656256696332292364270270137, 9.898991370242585629651535132176

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