Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 251 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.66·5-s − 1.81·7-s + 3.97·11-s − 4.88·13-s + 3.09·17-s − 3.71·19-s − 2.52·23-s − 2.22·25-s − 4.16·29-s + 4.17·31-s − 3.02·35-s + 11.0·37-s + 0.0987·41-s − 2.04·43-s − 0.469·47-s − 3.69·49-s + 3.85·53-s + 6.62·55-s − 7.09·59-s − 0.640·61-s − 8.14·65-s + 1.93·67-s − 0.867·71-s − 6.51·73-s − 7.22·77-s − 9.82·79-s − 12.6·83-s + ⋯
L(s)  = 1  + 0.745·5-s − 0.686·7-s + 1.19·11-s − 1.35·13-s + 0.750·17-s − 0.853·19-s − 0.525·23-s − 0.444·25-s − 0.772·29-s + 0.750·31-s − 0.512·35-s + 1.81·37-s + 0.0154·41-s − 0.312·43-s − 0.0684·47-s − 0.528·49-s + 0.530·53-s + 0.893·55-s − 0.924·59-s − 0.0819·61-s − 1.01·65-s + 0.236·67-s − 0.102·71-s − 0.762·73-s − 0.823·77-s − 1.10·79-s − 1.39·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9036\)    =    \(2^{2} \cdot 3^{2} \cdot 251\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;251\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
251 \( 1 + T \)
good5 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 - 3.97T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 - 3.09T + 17T^{2} \)
19 \( 1 + 3.71T + 19T^{2} \)
23 \( 1 + 2.52T + 23T^{2} \)
29 \( 1 + 4.16T + 29T^{2} \)
31 \( 1 - 4.17T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 0.0987T + 41T^{2} \)
43 \( 1 + 2.04T + 43T^{2} \)
47 \( 1 + 0.469T + 47T^{2} \)
53 \( 1 - 3.85T + 53T^{2} \)
59 \( 1 + 7.09T + 59T^{2} \)
61 \( 1 + 0.640T + 61T^{2} \)
67 \( 1 - 1.93T + 67T^{2} \)
71 \( 1 + 0.867T + 71T^{2} \)
73 \( 1 + 6.51T + 73T^{2} \)
79 \( 1 + 9.82T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 9.81T + 89T^{2} \)
97 \( 1 + 1.98T + 97T^{2} \)
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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

−7.39124468241116335086441025211, −6.51256729181908922924674171678, −6.16204388249107438811944362986, −5.45982575608307035939895007084, −4.52075841240327234931573540992, −3.91638074069008799804743133731, −2.94713444689768723966074668416, −2.22147712990958123545574776161, −1.32181619930289266960357592570, 0, 1.32181619930289266960357592570, 2.22147712990958123545574776161, 2.94713444689768723966074668416, 3.91638074069008799804743133731, 4.52075841240327234931573540992, 5.45982575608307035939895007084, 6.16204388249107438811944362986, 6.51256729181908922924674171678, 7.39124468241116335086441025211

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