Properties

Label 2-4008-1.1-c1-0-66
Degree $2$
Conductor $4008$
Sign $-1$
Analytic cond. $32.0040$
Root an. cond. $5.65721$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s + 9-s − 2·11-s + 2·13-s − 15-s − 21-s + 4·23-s − 4·25-s + 27-s − 6·29-s − 5·31-s − 2·33-s + 35-s − 37-s + 2·39-s + 2·41-s − 45-s − 47-s − 6·49-s + 9·53-s + 2·55-s − 5·59-s − 12·61-s − 63-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s − 0.218·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.348·33-s + 0.169·35-s − 0.164·37-s + 0.320·39-s + 0.312·41-s − 0.149·45-s − 0.145·47-s − 6/7·49-s + 1.23·53-s + 0.269·55-s − 0.650·59-s − 1.53·61-s − 0.125·63-s − 0.248·65-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: yes
Arithmetic: yes
Character: $\chi_{4008} (1, \cdot )$
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 \( 1 + T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−7.945106184052585139758379575266, −7.55308257744982315753758664140, −6.73311984688068827108858761641, −5.85129300030107946915719638810, −5.07098989305488583971338291559, −4.04530078291200321778071830236, −3.45053522856135647539257962566, −2.59211203970254782164506873562, −1.50704790034368395444312726686, 0, 1.50704790034368395444312726686, 2.59211203970254782164506873562, 3.45053522856135647539257962566, 4.04530078291200321778071830236, 5.07098989305488583971338291559, 5.85129300030107946915719638810, 6.73311984688068827108858761641, 7.55308257744982315753758664140, 7.945106184052585139758379575266

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