Properties

Label 2-8040-1.1-c1-0-76
Degree $2$
Conductor $8040$
Sign $1$
Analytic cond. $64.1997$
Root an. cond. $8.01247$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2.56·7-s + 9-s + 1.43·11-s + 5.12·13-s + 15-s + 1.12·17-s − 1.12·19-s + 2.56·21-s − 4·23-s + 25-s + 27-s + 7.12·29-s + 7.12·31-s + 1.43·33-s + 2.56·35-s − 5.68·37-s + 5.12·39-s + 6·41-s + 2.87·43-s + 45-s − 5.12·47-s − 0.438·49-s + 1.12·51-s + 3.12·53-s + 1.43·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.968·7-s + 0.333·9-s + 0.433·11-s + 1.42·13-s + 0.258·15-s + 0.272·17-s − 0.257·19-s + 0.558·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s + 1.32·29-s + 1.27·31-s + 0.250·33-s + 0.432·35-s − 0.934·37-s + 0.820·39-s + 0.937·41-s + 0.438·43-s + 0.149·45-s − 0.747·47-s − 0.0626·49-s + 0.157·51-s + 0.428·53-s + 0.193·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 67\)
Sign: $1$
Analytic conductor: \(64.1997\)
Root analytic conductor: \(8.01247\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.909860017\)
\(L(\frac12)\) \(\approx\) \(3.909860017\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
67 \( 1 - T \)
good7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 - 1.43T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 7.12T + 29T^{2} \)
31 \( 1 - 7.12T + 31T^{2} \)
37 \( 1 + 5.68T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 2.87T + 43T^{2} \)
47 \( 1 + 5.12T + 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 5.68T + 61T^{2} \)
71 \( 1 - 7.43T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 9.43T + 83T^{2} \)
89 \( 1 + 2.80T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−8.101997652440054771685517138884, −7.19424823207499878875399265354, −6.33997071603033240948723384780, −5.92492918633722996570413706114, −4.88933334625498304352646673370, −4.29508896480258978866115539792, −3.51183954763214654532974670335, −2.62996698887298128459456258773, −1.69509706841606224362470294959, −1.06311274097880722048902186751, 1.06311274097880722048902186751, 1.69509706841606224362470294959, 2.62996698887298128459456258773, 3.51183954763214654532974670335, 4.29508896480258978866115539792, 4.88933334625498304352646673370, 5.92492918633722996570413706114, 6.33997071603033240948723384780, 7.19424823207499878875399265354, 8.101997652440054771685517138884

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