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 − 0.302·7-s + 2.69·13-s − 2.30·17-s + 2·19-s + 2.30·23-s + 4·25-s − 7.60·29-s + 6.60·31-s − 0.908·35-s + 0.394·37-s + 6.21·41-s + 9.60·43-s − 1.60·47-s − 6.90·49-s + 4.60·53-s − 7.81·59-s + 6.81·61-s + 8.09·65-s + 8.21·67-s − 3.90·71-s − 3.09·73-s + 6.39·79-s + 1.60·83-s − 6.90·85-s + 13.8·89-s − 0.816·91-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.114·7-s + 0.748·13-s − 0.558·17-s + 0.458·19-s + 0.480·23-s + 0.800·25-s − 1.41·29-s + 1.18·31-s − 0.153·35-s + 0.0648·37-s + 0.970·41-s + 1.46·43-s − 0.234·47-s − 0.986·49-s + 0.632·53-s − 1.01·59-s + 0.872·61-s + 1.00·65-s + 1.00·67-s − 0.463·71-s − 0.361·73-s + 0.719·79-s + 0.176·83-s − 0.749·85-s + 1.46·89-s − 0.0856·91-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  =  $(2,\ 6012,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.794652604$
$L(\frac12)$  $\approx$  $2.794652604$
$L(\frac{3}{2})$   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 \( 1 + 0.302T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 + 2.30T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 2.30T + 23T^{2} \)
29 \( 1 + 7.60T + 29T^{2} \)
31 \( 1 - 6.60T + 31T^{2} \)
37 \( 1 - 0.394T + 37T^{2} \)
41 \( 1 - 6.21T + 41T^{2} \)
43 \( 1 - 9.60T + 43T^{2} \)
47 \( 1 + 1.60T + 47T^{2} \)
53 \( 1 - 4.60T + 53T^{2} \)
59 \( 1 + 7.81T + 59T^{2} \)
61 \( 1 - 6.81T + 61T^{2} \)
67 \( 1 - 8.21T + 67T^{2} \)
71 \( 1 + 3.90T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 - 6.39T + 79T^{2} \)
83 \( 1 - 1.60T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 4.51T + 97T^{2} \)
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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

−8.101739136351671933754826703925, −7.31744006795061071169621830091, −6.47718426162445350139061455084, −5.98879049735523798975252584910, −5.36762365329716937398940319490, −4.51976553544408990747927626230, −3.58959185030604405792808054236, −2.64463515503432810562383424323, −1.89342785702541234455822388234, −0.910994184126424918935185357619, 0.910994184126424918935185357619, 1.89342785702541234455822388234, 2.64463515503432810562383424323, 3.58959185030604405792808054236, 4.51976553544408990747927626230, 5.36762365329716937398940319490, 5.98879049735523798975252584910, 6.47718426162445350139061455084, 7.31744006795061071169621830091, 8.101739136351671933754826703925

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