Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.98·5-s − 0.0909·7-s + 9-s − 2.35·11-s + 0.834·13-s − 1.98·15-s + 7.08·17-s − 1.66·19-s − 0.0909·21-s − 2.49·23-s − 1.05·25-s + 27-s − 3.96·29-s − 3.52·31-s − 2.35·33-s + 0.180·35-s − 1.64·37-s + 0.834·39-s + 0.142·41-s − 5.36·43-s − 1.98·45-s − 6.35·47-s − 6.99·49-s + 7.08·51-s + 13.4·53-s + 4.67·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.888·5-s − 0.0343·7-s + 0.333·9-s − 0.710·11-s + 0.231·13-s − 0.512·15-s + 1.71·17-s − 0.382·19-s − 0.0198·21-s − 0.521·23-s − 0.211·25-s + 0.192·27-s − 0.736·29-s − 0.633·31-s − 0.410·33-s + 0.0305·35-s − 0.270·37-s + 0.133·39-s + 0.0223·41-s − 0.817·43-s − 0.296·45-s − 0.926·47-s − 0.998·49-s + 0.992·51-s + 1.84·53-s + 0.630·55-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  =  1
Selberg data  =  $(2,\ 4008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.98T + 5T^{2} \)
7 \( 1 + 0.0909T + 7T^{2} \)
11 \( 1 + 2.35T + 11T^{2} \)
13 \( 1 - 0.834T + 13T^{2} \)
17 \( 1 - 7.08T + 17T^{2} \)
19 \( 1 + 1.66T + 19T^{2} \)
23 \( 1 + 2.49T + 23T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 + 3.52T + 31T^{2} \)
37 \( 1 + 1.64T + 37T^{2} \)
41 \( 1 - 0.142T + 41T^{2} \)
43 \( 1 + 5.36T + 43T^{2} \)
47 \( 1 + 6.35T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 - 4.39T + 59T^{2} \)
61 \( 1 - 0.0766T + 61T^{2} \)
67 \( 1 - 4.51T + 67T^{2} \)
71 \( 1 - 3.17T + 71T^{2} \)
73 \( 1 - 9.30T + 73T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 - 0.0141T + 83T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.109158409378309537396991928880, −7.54126955541304936273479435675, −6.83950226391713457094170382330, −5.73385309615965575104891251543, −5.11272114179857448220479912907, −3.95407517375463636507934667037, −3.55678940592413695348604713581, −2.60225418646416617478139066579, −1.46454003648064016646555818374, 0, 1.46454003648064016646555818374, 2.60225418646416617478139066579, 3.55678940592413695348604713581, 3.95407517375463636507934667037, 5.11272114179857448220479912907, 5.73385309615965575104891251543, 6.83950226391713457094170382330, 7.54126955541304936273479435675, 8.109158409378309537396991928880

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