Properties

Label 2-4008-1.1-c1-0-56
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 + 0.392·5-s + 0.0476·7-s + 9-s + 1.67·11-s − 4.70·13-s − 0.392·15-s + 0.835·17-s + 0.998·19-s − 0.0476·21-s + 1.31·23-s − 4.84·25-s − 27-s + 6.86·29-s − 9.48·31-s − 1.67·33-s + 0.0187·35-s + 2.44·37-s + 4.70·39-s + 0.448·41-s + 9.51·43-s + 0.392·45-s − 3.80·47-s − 6.99·49-s − 0.835·51-s − 9.39·53-s + 0.658·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.175·5-s + 0.0180·7-s + 0.333·9-s + 0.505·11-s − 1.30·13-s − 0.101·15-s + 0.202·17-s + 0.229·19-s − 0.0104·21-s + 0.273·23-s − 0.969·25-s − 0.192·27-s + 1.27·29-s − 1.70·31-s − 0.291·33-s + 0.00316·35-s + 0.402·37-s + 0.753·39-s + 0.0699·41-s + 1.45·43-s + 0.0584·45-s − 0.555·47-s − 0.999·49-s − 0.117·51-s − 1.29·53-s + 0.0887·55-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 \( 1 - 0.392T + 5T^{2} \)
7 \( 1 - 0.0476T + 7T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 - 0.835T + 17T^{2} \)
19 \( 1 - 0.998T + 19T^{2} \)
23 \( 1 - 1.31T + 23T^{2} \)
29 \( 1 - 6.86T + 29T^{2} \)
31 \( 1 + 9.48T + 31T^{2} \)
37 \( 1 - 2.44T + 37T^{2} \)
41 \( 1 - 0.448T + 41T^{2} \)
43 \( 1 - 9.51T + 43T^{2} \)
47 \( 1 + 3.80T + 47T^{2} \)
53 \( 1 + 9.39T + 53T^{2} \)
59 \( 1 + 3.24T + 59T^{2} \)
61 \( 1 - 0.335T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 - 5.78T + 71T^{2} \)
73 \( 1 + 2.80T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 7.43T + 83T^{2} \)
89 \( 1 + 5.92T + 89T^{2} \)
97 \( 1 - 4.70T + 97T^{2} \)
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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

−7.85212753715718938985880732801, −7.39452373504383855911989698188, −6.54202570519821928879447452429, −5.88474848517432986207048160920, −5.06524406266577019100882681222, −4.44656581523703015876617136656, −3.44482222992153179654229613831, −2.40527645155454940802793754072, −1.36578187953052818518468729009, 0, 1.36578187953052818518468729009, 2.40527645155454940802793754072, 3.44482222992153179654229613831, 4.44656581523703015876617136656, 5.06524406266577019100882681222, 5.88474848517432986207048160920, 6.54202570519821928879447452429, 7.39452373504383855911989698188, 7.85212753715718938985880732801

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