Properties

Label 2-4008-1.1-c1-0-38
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 − 4.44·5-s + 2.42·7-s + 9-s − 0.762·11-s + 0.148·13-s + 4.44·15-s − 0.334·17-s − 4.24·19-s − 2.42·21-s + 4.31·23-s + 14.7·25-s − 27-s − 7.41·29-s − 2.55·31-s + 0.762·33-s − 10.7·35-s + 10.9·37-s − 0.148·39-s + 2.59·41-s + 4.61·43-s − 4.44·45-s − 3.82·47-s − 1.10·49-s + 0.334·51-s + 10.1·53-s + 3.39·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.98·5-s + 0.917·7-s + 0.333·9-s − 0.229·11-s + 0.0413·13-s + 1.14·15-s − 0.0811·17-s − 0.973·19-s − 0.529·21-s + 0.900·23-s + 2.95·25-s − 0.192·27-s − 1.37·29-s − 0.458·31-s + 0.132·33-s − 1.82·35-s + 1.79·37-s − 0.0238·39-s + 0.404·41-s + 0.704·43-s − 0.662·45-s − 0.558·47-s − 0.158·49-s + 0.0468·51-s + 1.39·53-s + 0.457·55-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 \( 1 + 4.44T + 5T^{2} \)
7 \( 1 - 2.42T + 7T^{2} \)
11 \( 1 + 0.762T + 11T^{2} \)
13 \( 1 - 0.148T + 13T^{2} \)
17 \( 1 + 0.334T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 - 4.31T + 23T^{2} \)
29 \( 1 + 7.41T + 29T^{2} \)
31 \( 1 + 2.55T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 2.59T + 41T^{2} \)
43 \( 1 - 4.61T + 43T^{2} \)
47 \( 1 + 3.82T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 - 6.52T + 59T^{2} \)
61 \( 1 - 0.628T + 61T^{2} \)
67 \( 1 + 6.47T + 67T^{2} \)
71 \( 1 + 0.134T + 71T^{2} \)
73 \( 1 - 3.02T + 73T^{2} \)
79 \( 1 + 1.69T + 79T^{2} \)
83 \( 1 + 4.64T + 83T^{2} \)
89 \( 1 + 9.98T + 89T^{2} \)
97 \( 1 - 9.08T + 97T^{2} \)
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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

−7.992434411447022878320774814751, −7.42460118370920994774168961337, −6.85473468651847566940712448611, −5.76123690036235351256007987656, −4.88536906818550079205797535311, −4.29235733326155285401589694527, −3.70889619459356795876082338284, −2.53469550356192433916511095542, −1.10888687384730257824617762195, 0, 1.10888687384730257824617762195, 2.53469550356192433916511095542, 3.70889619459356795876082338284, 4.29235733326155285401589694527, 4.88536906818550079205797535311, 5.76123690036235351256007987656, 6.85473468651847566940712448611, 7.42460118370920994774168961337, 7.992434411447022878320774814751

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