Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \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  + 3·5-s + 3.30·7-s + 6.30·13-s + 1.30·17-s + 2·19-s − 1.30·23-s + 4·25-s − 0.394·29-s − 0.605·31-s + 9.90·35-s + 7.60·37-s − 8.21·41-s + 2.39·43-s + 5.60·47-s + 3.90·49-s − 2.60·53-s + 13.8·59-s − 14.8·61-s + 18.9·65-s − 6.21·67-s + 6.90·71-s − 13.9·73-s + 13.6·79-s − 5.60·83-s + 3.90·85-s − 7.81·89-s + 20.8·91-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.24·7-s + 1.74·13-s + 0.315·17-s + 0.458·19-s − 0.271·23-s + 0.800·25-s − 0.0732·29-s − 0.108·31-s + 1.67·35-s + 1.25·37-s − 1.28·41-s + 0.365·43-s + 0.817·47-s + 0.558·49-s − 0.357·53-s + 1.79·59-s − 1.89·61-s + 2.34·65-s − 0.758·67-s + 0.819·71-s − 1.62·73-s + 1.53·79-s − 0.615·83-s + 0.423·85-s − 0.828·89-s + 2.18·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6012,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.672965610$
$L(\frac12)$  $\approx$  $3.672965610$
$L(\frac{3}{2})$   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 \( 1 \)
167 \( 1 - T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.30T + 13T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 1.30T + 23T^{2} \)
29 \( 1 + 0.394T + 29T^{2} \)
31 \( 1 + 0.605T + 31T^{2} \)
37 \( 1 - 7.60T + 37T^{2} \)
41 \( 1 + 8.21T + 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 - 5.60T + 47T^{2} \)
53 \( 1 + 2.60T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 6.21T + 67T^{2} \)
71 \( 1 - 6.90T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 5.60T + 83T^{2} \)
89 \( 1 + 7.81T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−8.187282186472679021982844735090, −7.44346826825739842210961240083, −6.47512391492057174641361757289, −5.85936947209048348045760151659, −5.39367056971665751879451724818, −4.53044945036982782249640413760, −3.67165279116746009034227060832, −2.62723449833689006743733027752, −1.66328674420939585085062941828, −1.17500671843368084589159178289, 1.17500671843368084589159178289, 1.66328674420939585085062941828, 2.62723449833689006743733027752, 3.67165279116746009034227060832, 4.53044945036982782249640413760, 5.39367056971665751879451724818, 5.85936947209048348045760151659, 6.47512391492057174641361757289, 7.44346826825739842210961240083, 8.187282186472679021982844735090

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