Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.49·5-s + 4.58·7-s + 9-s + 0.512·11-s − 0.325·13-s − 3.49·15-s − 2.49·17-s + 3.26·19-s + 4.58·21-s − 4.00·23-s + 7.20·25-s + 27-s + 1.72·29-s + 5.79·31-s + 0.512·33-s − 16.0·35-s + 7.03·37-s − 0.325·39-s + 1.69·41-s − 7.20·43-s − 3.49·45-s + 2.49·47-s + 13.9·49-s − 2.49·51-s + 2.26·53-s − 1.79·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.56·5-s + 1.73·7-s + 0.333·9-s + 0.154·11-s − 0.0903·13-s − 0.902·15-s − 0.604·17-s + 0.748·19-s + 0.999·21-s − 0.835·23-s + 1.44·25-s + 0.192·27-s + 0.320·29-s + 1.04·31-s + 0.0892·33-s − 2.70·35-s + 1.15·37-s − 0.0521·39-s + 0.264·41-s − 1.09·43-s − 0.520·45-s + 0.363·47-s + 1.99·49-s − 0.348·51-s + 0.311·53-s − 0.241·55-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

\( d \)  =  \(2\)
\( N \)  =  \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.188343758$
$L(\frac12)$  $\approx$  $2.188343758$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 + 3.49T + 5T^{2} \)
7 \( 1 - 4.58T + 7T^{2} \)
11 \( 1 - 0.512T + 11T^{2} \)
13 \( 1 + 0.325T + 13T^{2} \)
17 \( 1 + 2.49T + 17T^{2} \)
19 \( 1 - 3.26T + 19T^{2} \)
23 \( 1 + 4.00T + 23T^{2} \)
29 \( 1 - 1.72T + 29T^{2} \)
31 \( 1 - 5.79T + 31T^{2} \)
37 \( 1 - 7.03T + 37T^{2} \)
41 \( 1 - 1.69T + 41T^{2} \)
43 \( 1 + 7.20T + 43T^{2} \)
47 \( 1 - 2.49T + 47T^{2} \)
53 \( 1 - 2.26T + 53T^{2} \)
59 \( 1 + 6.76T + 59T^{2} \)
61 \( 1 + 1.57T + 61T^{2} \)
67 \( 1 - 0.691T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 3.41T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 - 8.68T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.275882514209447165354293198353, −7.82120678160411716862227818125, −7.38097750240993845002788003482, −6.39582623992654996617191617647, −5.16007325984587631042295133015, −4.48041503400021122149954588325, −4.02841736362897477494913093650, −3.02425665992598589853575493818, −1.97854773439203712344921041907, −0.853361803869348927451257825855, 0.853361803869348927451257825855, 1.97854773439203712344921041907, 3.02425665992598589853575493818, 4.02841736362897477494913093650, 4.48041503400021122149954588325, 5.16007325984587631042295133015, 6.39582623992654996617191617647, 7.38097750240993845002788003482, 7.82120678160411716862227818125, 8.275882514209447165354293198353

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