Properties

Label 2-4008-1.1-c1-0-30
Degree $2$
Conductor $4008$
Sign $-1$
Analytic cond. $32.0040$
Root an. cond. $5.65721$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.24·5-s − 3.21·7-s + 9-s + 2.82·11-s − 5.85·13-s + 3.24·15-s + 1.86·17-s + 5.37·19-s + 3.21·21-s + 4.21·23-s + 5.51·25-s − 27-s + 3.33·29-s + 1.27·31-s − 2.82·33-s + 10.4·35-s − 2.06·37-s + 5.85·39-s + 2.10·41-s + 6.96·43-s − 3.24·45-s − 6.68·47-s + 3.30·49-s − 1.86·51-s + 9.34·53-s − 9.14·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.45·5-s − 1.21·7-s + 0.333·9-s + 0.850·11-s − 1.62·13-s + 0.837·15-s + 0.451·17-s + 1.23·19-s + 0.700·21-s + 0.877·23-s + 1.10·25-s − 0.192·27-s + 0.618·29-s + 0.228·31-s − 0.491·33-s + 1.75·35-s − 0.339·37-s + 0.936·39-s + 0.329·41-s + 1.06·43-s − 0.483·45-s − 0.975·47-s + 0.472·49-s − 0.260·51-s + 1.28·53-s − 1.23·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-1$
Analytic conductor: \(32.0040\)
Root analytic conductor: \(5.65721\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4008,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 + 3.24T + 5T^{2} \)
7 \( 1 + 3.21T + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 - 1.86T + 17T^{2} \)
19 \( 1 - 5.37T + 19T^{2} \)
23 \( 1 - 4.21T + 23T^{2} \)
29 \( 1 - 3.33T + 29T^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 + 2.06T + 37T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 + 6.68T + 47T^{2} \)
53 \( 1 - 9.34T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 7.78T + 61T^{2} \)
67 \( 1 - 3.51T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 + 4.16T + 79T^{2} \)
83 \( 1 + 1.27T + 83T^{2} \)
89 \( 1 + 5.91T + 89T^{2} \)
97 \( 1 - 4.01T + 97T^{2} \)
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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

−7.85526946803841754792531138452, −7.19371947921875298803968996527, −6.88169182171200331354819980607, −5.88534184852384024657623811823, −4.98472078179642878453907895383, −4.27363998691511720042420701841, −3.42677472917950704605935964772, −2.78604943467038034466762795978, −1.01087658527285487659632088287, 0, 1.01087658527285487659632088287, 2.78604943467038034466762795978, 3.42677472917950704605935964772, 4.27363998691511720042420701841, 4.98472078179642878453907895383, 5.88534184852384024657623811823, 6.88169182171200331354819980607, 7.19371947921875298803968996527, 7.85526946803841754792531138452

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