Properties

Label 2-504-1.1-c3-0-12
Degree $2$
Conductor $504$
Sign $-1$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20.3·5-s + 7·7-s + 30.9·11-s + 50.6·13-s + 102.·17-s − 61.2·19-s − 148.·23-s + 287.·25-s − 159.·29-s − 121.·31-s − 142.·35-s − 357.·37-s − 466.·41-s − 185.·43-s + 131.·47-s + 49·49-s − 200.·53-s − 627.·55-s + 591.·59-s + 70.5·61-s − 1.02e3·65-s − 643.·67-s + 522.·71-s − 576.·73-s + 216.·77-s + 280.·79-s + 557.·83-s + ⋯
L(s)  = 1  − 1.81·5-s + 0.377·7-s + 0.847·11-s + 1.07·13-s + 1.46·17-s − 0.739·19-s − 1.34·23-s + 2.29·25-s − 1.01·29-s − 0.702·31-s − 0.686·35-s − 1.59·37-s − 1.77·41-s − 0.658·43-s + 0.407·47-s + 0.142·49-s − 0.518·53-s − 1.53·55-s + 1.30·59-s + 0.148·61-s − 1.96·65-s − 1.17·67-s + 0.873·71-s − 0.923·73-s + 0.320·77-s + 0.399·79-s + 0.737·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 7T \)
good5 \( 1 + 20.3T + 125T^{2} \)
11 \( 1 - 30.9T + 1.33e3T^{2} \)
13 \( 1 - 50.6T + 2.19e3T^{2} \)
17 \( 1 - 102.T + 4.91e3T^{2} \)
19 \( 1 + 61.2T + 6.85e3T^{2} \)
23 \( 1 + 148.T + 1.21e4T^{2} \)
29 \( 1 + 159.T + 2.43e4T^{2} \)
31 \( 1 + 121.T + 2.97e4T^{2} \)
37 \( 1 + 357.T + 5.06e4T^{2} \)
41 \( 1 + 466.T + 6.89e4T^{2} \)
43 \( 1 + 185.T + 7.95e4T^{2} \)
47 \( 1 - 131.T + 1.03e5T^{2} \)
53 \( 1 + 200.T + 1.48e5T^{2} \)
59 \( 1 - 591.T + 2.05e5T^{2} \)
61 \( 1 - 70.5T + 2.26e5T^{2} \)
67 \( 1 + 643.T + 3.00e5T^{2} \)
71 \( 1 - 522.T + 3.57e5T^{2} \)
73 \( 1 + 576.T + 3.89e5T^{2} \)
79 \( 1 - 280.T + 4.93e5T^{2} \)
83 \( 1 - 557.T + 5.71e5T^{2} \)
89 \( 1 - 1.22e3T + 7.04e5T^{2} \)
97 \( 1 - 65.0T + 9.12e5T^{2} \)
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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

−10.25766199974540252842845590230, −8.872034337651921528095099355461, −8.227884692787713696896016997207, −7.50628729151117848897611452819, −6.50140004726001976394573395993, −5.21944881798502008165559737448, −3.83930087190104098589088187740, −3.62111958497446769258211289943, −1.48465324693089921411677507212, 0, 1.48465324693089921411677507212, 3.62111958497446769258211289943, 3.83930087190104098589088187740, 5.21944881798502008165559737448, 6.50140004726001976394573395993, 7.50628729151117848897611452819, 8.227884692787713696896016997207, 8.872034337651921528095099355461, 10.25766199974540252842845590230

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