Properties

Degree 2
Conductor 3
Sign $-1$
Motivic weight 43
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.56e6·2-s + 1.04e10·3-s + 1.20e13·4-s − 1.29e15·5-s + 4.77e16·6-s − 9.66e17·7-s + 1.47e19·8-s + 1.09e20·9-s − 5.89e21·10-s − 4.14e22·11-s + 1.25e23·12-s + 6.85e23·13-s − 4.41e24·14-s − 1.35e25·15-s − 3.86e25·16-s − 1.87e26·17-s + 4.99e26·18-s − 3.67e27·19-s − 1.55e28·20-s − 1.01e28·21-s − 1.89e29·22-s + 2.38e29·23-s + 1.53e29·24-s + 5.33e29·25-s + 3.12e30·26-s + 1.14e30·27-s − 1.16e31·28-s + ⋯
L(s)  = 1  + 1.53·2-s + 0.577·3-s + 1.36·4-s − 1.21·5-s + 0.888·6-s − 0.654·7-s + 0.564·8-s + 0.333·9-s − 1.86·10-s − 1.69·11-s + 0.789·12-s + 0.769·13-s − 1.00·14-s − 0.699·15-s − 0.498·16-s − 0.659·17-s + 0.512·18-s − 1.18·19-s − 1.65·20-s − 0.377·21-s − 2.60·22-s + 1.25·23-s + 0.325·24-s + 0.468·25-s + 1.18·26-s + 0.192·27-s − 0.894·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(43\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 3,\ (\ :43/2),\ -1)\)
\(L(22)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{45}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 1.04e10T \)
good2 \( 1 - 4.56e6T + 8.79e12T^{2} \)
5 \( 1 + 1.29e15T + 1.13e30T^{2} \)
7 \( 1 + 9.66e17T + 2.18e36T^{2} \)
11 \( 1 + 4.14e22T + 6.02e44T^{2} \)
13 \( 1 - 6.85e23T + 7.93e47T^{2} \)
17 \( 1 + 1.87e26T + 8.11e52T^{2} \)
19 \( 1 + 3.67e27T + 9.69e54T^{2} \)
23 \( 1 - 2.38e29T + 3.58e58T^{2} \)
29 \( 1 - 4.15e31T + 7.64e62T^{2} \)
31 \( 1 + 3.15e31T + 1.34e64T^{2} \)
37 \( 1 + 6.10e32T + 2.70e67T^{2} \)
41 \( 1 + 4.47e34T + 2.23e69T^{2} \)
43 \( 1 + 2.07e35T + 1.73e70T^{2} \)
47 \( 1 - 1.73e36T + 7.94e71T^{2} \)
53 \( 1 + 1.34e37T + 1.39e74T^{2} \)
59 \( 1 + 7.90e37T + 1.40e76T^{2} \)
61 \( 1 - 1.04e38T + 5.87e76T^{2} \)
67 \( 1 - 2.71e39T + 3.32e78T^{2} \)
71 \( 1 - 6.41e39T + 4.01e79T^{2} \)
73 \( 1 - 4.03e39T + 1.32e80T^{2} \)
79 \( 1 + 6.03e40T + 3.96e81T^{2} \)
83 \( 1 - 2.83e41T + 3.31e82T^{2} \)
89 \( 1 + 7.96e41T + 6.66e83T^{2} \)
97 \( 1 + 2.28e41T + 2.69e85T^{2} \)
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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

−15.26722349363469605233869927055, −13.47258992607920183743076766243, −12.57588880279262339539561938014, −10.89221785549526099792673120290, −8.358323044125470451079358831124, −6.70818696674847612958340526260, −4.85715559143026608315069480264, −3.62575963961869245448965658226, −2.63625675068958808595944468931, 0, 2.63625675068958808595944468931, 3.62575963961869245448965658226, 4.85715559143026608315069480264, 6.70818696674847612958340526260, 8.358323044125470451079358831124, 10.89221785549526099792673120290, 12.57588880279262339539561938014, 13.47258992607920183743076766243, 15.26722349363469605233869927055

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