Properties

Label 2-4008-1.1-c1-0-59
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 + 2.45·5-s − 0.714·7-s + 9-s − 1.07·11-s − 6.69·13-s − 2.45·15-s + 1.59·17-s + 5.29·19-s + 0.714·21-s + 0.914·23-s + 1.04·25-s − 27-s − 8.70·29-s + 7.73·31-s + 1.07·33-s − 1.75·35-s − 5.65·37-s + 6.69·39-s + 5.51·41-s − 6.28·43-s + 2.45·45-s + 0.0858·47-s − 6.49·49-s − 1.59·51-s − 0.447·53-s − 2.65·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.09·5-s − 0.269·7-s + 0.333·9-s − 0.325·11-s − 1.85·13-s − 0.635·15-s + 0.387·17-s + 1.21·19-s + 0.155·21-s + 0.190·23-s + 0.209·25-s − 0.192·27-s − 1.61·29-s + 1.38·31-s + 0.187·33-s − 0.296·35-s − 0.928·37-s + 1.07·39-s + 0.860·41-s − 0.958·43-s + 0.366·45-s + 0.0125·47-s − 0.927·49-s − 0.223·51-s − 0.0614·53-s − 0.357·55-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 \( 1 - 2.45T + 5T^{2} \)
7 \( 1 + 0.714T + 7T^{2} \)
11 \( 1 + 1.07T + 11T^{2} \)
13 \( 1 + 6.69T + 13T^{2} \)
17 \( 1 - 1.59T + 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 - 0.914T + 23T^{2} \)
29 \( 1 + 8.70T + 29T^{2} \)
31 \( 1 - 7.73T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 5.51T + 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 - 0.0858T + 47T^{2} \)
53 \( 1 + 0.447T + 53T^{2} \)
59 \( 1 - 4.05T + 59T^{2} \)
61 \( 1 + 5.73T + 61T^{2} \)
67 \( 1 + 9.03T + 67T^{2} \)
71 \( 1 - 0.319T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 2.88T + 83T^{2} \)
89 \( 1 + 5.37T + 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−7.85137601253887424224873944555, −7.31773623336193446882334887589, −6.57151778625119773677291830421, −5.68610189175874541039506036351, −5.27084928001390621451734737313, −4.54789094518474858259798614971, −3.24780639095206052665322855110, −2.42554632453823099519741862492, −1.45104423453994539580849783908, 0, 1.45104423453994539580849783908, 2.42554632453823099519741862492, 3.24780639095206052665322855110, 4.54789094518474858259798614971, 5.27084928001390621451734737313, 5.68610189175874541039506036351, 6.57151778625119773677291830421, 7.31773623336193446882334887589, 7.85137601253887424224873944555

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