Properties

Label 2-2520-1.1-c1-0-8
Degree $2$
Conductor $2520$
Sign $1$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
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  − 5-s + 7-s + 5·11-s − 5·13-s + 7·17-s − 2·19-s + 2·23-s + 25-s − 7·29-s + 4·31-s − 35-s − 6·37-s + 12·41-s − 2·43-s − 47-s + 49-s − 5·55-s + 4·59-s + 4·61-s + 5·65-s + 8·67-s + 6·73-s + 5·77-s − 3·79-s + 4·83-s − 7·85-s − 5·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.50·11-s − 1.38·13-s + 1.69·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.29·29-s + 0.718·31-s − 0.169·35-s − 0.986·37-s + 1.87·41-s − 0.304·43-s − 0.145·47-s + 1/7·49-s − 0.674·55-s + 0.520·59-s + 0.512·61-s + 0.620·65-s + 0.977·67-s + 0.702·73-s + 0.569·77-s − 0.337·79-s + 0.439·83-s − 0.759·85-s − 0.524·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.831603530\)
\(L(\frac12)\) \(\approx\) \(1.831603530\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−8.986591679604333601542857191321, −8.033082332859894178672793368914, −7.42695240476731979776224139494, −6.74907559299019305057473208578, −5.74978286836860489420343080410, −4.93936350214085093868248089925, −4.06954744861732417689986542370, −3.30527145306019394907962235588, −2.07628009905382843609630348286, −0.887538960576224120433585893779, 0.887538960576224120433585893779, 2.07628009905382843609630348286, 3.30527145306019394907962235588, 4.06954744861732417689986542370, 4.93936350214085093868248089925, 5.74978286836860489420343080410, 6.74907559299019305057473208578, 7.42695240476731979776224139494, 8.033082332859894178672793368914, 8.986591679604333601542857191321

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