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 − 0.619·5-s + 0.954·7-s + 9-s + 6.25·11-s + 2.92·13-s − 0.619·15-s + 5.33·17-s + 2.94·19-s + 0.954·21-s − 2.33·23-s − 4.61·25-s + 27-s + 4.47·29-s − 3.45·31-s + 6.25·33-s − 0.590·35-s + 4.26·37-s + 2.92·39-s − 10.8·41-s − 1.32·43-s − 0.619·45-s + 3.37·47-s − 6.08·49-s + 5.33·51-s + 5.72·53-s − 3.87·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.276·5-s + 0.360·7-s + 0.333·9-s + 1.88·11-s + 0.811·13-s − 0.159·15-s + 1.29·17-s + 0.675·19-s + 0.208·21-s − 0.487·23-s − 0.923·25-s + 0.192·27-s + 0.830·29-s − 0.620·31-s + 1.08·33-s − 0.0998·35-s + 0.701·37-s + 0.468·39-s − 1.68·41-s − 0.201·43-s − 0.0923·45-s + 0.491·47-s − 0.869·49-s + 0.747·51-s + 0.786·53-s − 0.522·55-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  =  $(2,\ 4008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.017242044$
$L(\frac12)$  $\approx$  $3.017242044$
$L(\frac{3}{2})$   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 \( 1 + 0.619T + 5T^{2} \)
7 \( 1 - 0.954T + 7T^{2} \)
11 \( 1 - 6.25T + 11T^{2} \)
13 \( 1 - 2.92T + 13T^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
19 \( 1 - 2.94T + 19T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 - 4.26T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 - 3.37T + 47T^{2} \)
53 \( 1 - 5.72T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 0.604T + 67T^{2} \)
71 \( 1 + 9.74T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 - 0.00272T + 83T^{2} \)
89 \( 1 - 9.85T + 89T^{2} \)
97 \( 1 + 8.92T + 97T^{2} \)
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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

−8.418287063498549629391656703945, −7.83624951897204112652879992861, −7.05687353359578761993534483911, −6.29256462131453708789440495676, −5.54673206307631952766946580353, −4.44840330873279896252549398171, −3.73183144566666042558635631024, −3.22140961854196545189341851937, −1.78361052107674969049509889923, −1.09130026746448953081455861403, 1.09130026746448953081455861403, 1.78361052107674969049509889923, 3.22140961854196545189341851937, 3.73183144566666042558635631024, 4.44840330873279896252549398171, 5.54673206307631952766946580353, 6.29256462131453708789440495676, 7.05687353359578761993534483911, 7.83624951897204112652879992861, 8.418287063498549629391656703945

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