Properties

Degree 2
Conductor 3
Sign $1$
Motivic weight 6
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·3-s + 64·4-s − 286·7-s + 729·9-s − 1.72e3·12-s + 506·13-s + 4.09e3·16-s − 1.05e4·19-s + 7.72e3·21-s + 1.56e4·25-s − 1.96e4·27-s − 1.83e4·28-s + 3.52e4·31-s + 4.66e4·36-s − 8.92e4·37-s − 1.36e4·39-s + 1.11e5·43-s − 1.10e5·48-s − 3.58e4·49-s + 3.23e4·52-s + 2.85e5·57-s − 4.20e5·61-s − 2.08e5·63-s + 2.62e5·64-s + 1.72e5·67-s + 6.38e5·73-s − 4.21e5·75-s + ⋯
L(s)  = 1  − 3-s + 4-s − 0.833·7-s + 9-s − 12-s + 0.230·13-s + 16-s − 1.54·19-s + 0.833·21-s + 25-s − 27-s − 0.833·28-s + 1.18·31-s + 36-s − 1.76·37-s − 0.230·39-s + 1.40·43-s − 48-s − 0.304·49-s + 0.230·52-s + 1.54·57-s − 1.85·61-s − 0.833·63-s + 64-s + 0.574·67-s + 1.64·73-s − 75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(6\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :3),\ 1)$
$L(\frac{7}{2})$  $\approx$  $0.802526$
$L(\frac12)$  $\approx$  $0.802526$
$L(4)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + p^{3} T \)
good2 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
5 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
7 \( 1 + 286 T + p^{6} T^{2} \)
11 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
13 \( 1 - 506 T + p^{6} T^{2} \)
17 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
19 \( 1 + 10582 T + p^{6} T^{2} \)
23 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
29 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
31 \( 1 - 35282 T + p^{6} T^{2} \)
37 \( 1 + 89206 T + p^{6} T^{2} \)
41 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
43 \( 1 - 111386 T + p^{6} T^{2} \)
47 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
53 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
59 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
61 \( 1 + 420838 T + p^{6} T^{2} \)
67 \( 1 - 172874 T + p^{6} T^{2} \)
71 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
73 \( 1 - 638066 T + p^{6} T^{2} \)
79 \( 1 + 204622 T + p^{6} T^{2} \)
83 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
89 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
97 \( 1 + 56446 T + p^{6} T^{2} \)
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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

−25.91719154013093344414424320834, −24.31625596819603239877660545934, −22.81040584989870861590484456027, −21.14553963209100554789272819114, −19.16808984714065757854822219386, −16.98315802839373498452012617625, −15.64955611967823946795560751404, −12.47129821992859305641569206852, −10.64402516981818544954724445589, −6.51020658700370227865993577833, 6.51020658700370227865993577833, 10.64402516981818544954724445589, 12.47129821992859305641569206852, 15.64955611967823946795560751404, 16.98315802839373498452012617625, 19.16808984714065757854822219386, 21.14553963209100554789272819114, 22.81040584989870861590484456027, 24.31625596819603239877660545934, 25.91719154013093344414424320834

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