Properties

Label 2-5040-1.1-c1-0-3
Degree $2$
Conductor $5040$
Sign $1$
Analytic cond. $40.2446$
Root an. cond. $6.34386$
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 − 4·13-s − 6·17-s − 2·19-s + 6·23-s + 25-s − 2·31-s + 35-s + 2·37-s − 6·41-s + 4·43-s + 49-s + 6·53-s + 12·59-s − 10·61-s + 4·65-s + 4·67-s + 12·71-s − 4·73-s − 8·79-s + 12·83-s + 6·85-s − 6·89-s + 4·91-s + 2·95-s + 8·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.359·31-s + 0.169·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 0.488·67-s + 1.42·71-s − 0.468·73-s − 0.900·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s + 0.419·91-s + 0.205·95-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.105415283\)
\(L(\frac12)\) \(\approx\) \(1.105415283\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.312417494085986509106620907657, −7.32299067164021900893988658054, −6.95690613180478001261141513162, −6.19263020239855182713744341321, −5.16406092827354928586251405384, −4.58279289915766210559272218207, −3.76718008701211949698418282941, −2.80701696291884140593044580997, −2.05631859112990099961601757225, −0.54790171512367851136938199577, 0.54790171512367851136938199577, 2.05631859112990099961601757225, 2.80701696291884140593044580997, 3.76718008701211949698418282941, 4.58279289915766210559272218207, 5.16406092827354928586251405384, 6.19263020239855182713744341321, 6.95690613180478001261141513162, 7.32299067164021900893988658054, 8.312417494085986509106620907657

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