Properties

Label 2-6720-1.1-c1-0-37
Degree $2$
Conductor $6720$
Sign $1$
Analytic cond. $53.6594$
Root an. cond. $7.32526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 9-s + 2·11-s + 4·13-s − 15-s − 2·17-s − 6·19-s + 21-s + 4·23-s + 25-s + 27-s + 10·29-s − 2·31-s + 2·33-s − 35-s + 2·37-s + 4·39-s − 10·41-s + 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s − 4·53-s − 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s − 0.485·17-s − 1.37·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.359·31-s + 0.348·33-s − 0.169·35-s + 0.328·37-s + 0.640·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6720\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(53.6594\)
Root analytic conductor: \(7.32526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.693049733\)
\(L(\frac12)\) \(\approx\) \(2.693049733\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.202895607397633921034815918042, −7.27253775033112195641491208651, −6.63531859643468812020134360532, −6.06167445052925301847722751273, −4.91405803997892797007361916701, −4.28814657639165362498759127403, −3.65681516562296852641553769336, −2.78478654312962465632596899291, −1.82765064622844198050247586509, −0.852165152175321061071138373084, 0.852165152175321061071138373084, 1.82765064622844198050247586509, 2.78478654312962465632596899291, 3.65681516562296852641553769336, 4.28814657639165362498759127403, 4.91405803997892797007361916701, 6.06167445052925301847722751273, 6.63531859643468812020134360532, 7.27253775033112195641491208651, 8.202895607397633921034815918042

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