Properties

Label 2-4008-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4.02·5-s + 0.910·7-s + 9-s − 5.64·11-s + 6.43·13-s − 4.02·15-s + 3.30·17-s + 1.28·19-s + 0.910·21-s − 9.24·23-s + 11.2·25-s + 27-s + 2.90·29-s − 10.6·31-s − 5.64·33-s − 3.66·35-s + 3.18·37-s + 6.43·39-s + 1.79·41-s + 2.55·43-s − 4.02·45-s + 1.91·47-s − 6.17·49-s + 3.30·51-s − 6.55·53-s + 22.7·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.80·5-s + 0.343·7-s + 0.333·9-s − 1.70·11-s + 1.78·13-s − 1.03·15-s + 0.802·17-s + 0.295·19-s + 0.198·21-s − 1.92·23-s + 2.24·25-s + 0.192·27-s + 0.540·29-s − 1.90·31-s − 0.983·33-s − 0.619·35-s + 0.524·37-s + 1.03·39-s + 0.279·41-s + 0.390·43-s − 0.600·45-s + 0.279·47-s − 0.881·49-s + 0.463·51-s − 0.900·53-s + 3.06·55-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: \(0\)
Selberg data: \((2,\ 4008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428742949\)
\(L(\frac12)\) \(\approx\) \(1.428742949\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.02T + 5T^{2} \)
7 \( 1 - 0.910T + 7T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
13 \( 1 - 6.43T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 - 1.28T + 19T^{2} \)
23 \( 1 + 9.24T + 23T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 - 3.18T + 37T^{2} \)
41 \( 1 - 1.79T + 41T^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 - 1.91T + 47T^{2} \)
53 \( 1 + 6.55T + 53T^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 - 8.34T + 61T^{2} \)
67 \( 1 - 4.23T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 1.27T + 73T^{2} \)
79 \( 1 + 1.79T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 - 5.45T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−8.185639345401038531662998596355, −7.85013234515732036392503633765, −7.45001588280867255004461854415, −6.25859296324302249503834195622, −5.38031175739146304717477903657, −4.48394844904639316225010823117, −3.63762400968871490793783087844, −3.31259628720195122789434088064, −2.04427936880115221014454459412, −0.65695667694793754843734505187, 0.65695667694793754843734505187, 2.04427936880115221014454459412, 3.31259628720195122789434088064, 3.63762400968871490793783087844, 4.48394844904639316225010823117, 5.38031175739146304717477903657, 6.25859296324302249503834195622, 7.45001588280867255004461854415, 7.85013234515732036392503633765, 8.185639345401038531662998596355

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