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  + 1.51e6·2-s + 1.04e10·3-s − 6.49e12·4-s + 1.66e15·5-s + 1.58e16·6-s − 2.14e18·7-s − 2.31e19·8-s + 1.09e20·9-s + 2.52e21·10-s + 1.22e22·11-s − 6.79e22·12-s − 1.24e24·13-s − 3.25e24·14-s + 1.74e25·15-s + 2.20e25·16-s − 3.24e26·17-s + 1.65e26·18-s + 1.94e27·19-s − 1.08e28·20-s − 2.24e28·21-s + 1.86e28·22-s − 1.30e28·23-s − 2.42e29·24-s + 1.64e30·25-s − 1.88e30·26-s + 1.14e30·27-s + 1.39e31·28-s + ⋯
L(s)  = 1  + 0.511·2-s + 0.577·3-s − 0.738·4-s + 1.56·5-s + 0.295·6-s − 1.45·7-s − 0.888·8-s + 0.333·9-s + 0.799·10-s + 0.500·11-s − 0.426·12-s − 1.39·13-s − 0.742·14-s + 0.903·15-s + 0.284·16-s − 1.13·17-s + 0.170·18-s + 0.625·19-s − 1.15·20-s − 0.838·21-s + 0.255·22-s − 0.0690·23-s − 0.513·24-s + 1.44·25-s − 0.713·26-s + 0.192·27-s + 1.07·28-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 \( 1 - 1.51e6T + 8.79e12T^{2} \)
5 \( 1 - 1.66e15T + 1.13e30T^{2} \)
7 \( 1 + 2.14e18T + 2.18e36T^{2} \)
11 \( 1 - 1.22e22T + 6.02e44T^{2} \)
13 \( 1 + 1.24e24T + 7.93e47T^{2} \)
17 \( 1 + 3.24e26T + 8.11e52T^{2} \)
19 \( 1 - 1.94e27T + 9.69e54T^{2} \)
23 \( 1 + 1.30e28T + 3.58e58T^{2} \)
29 \( 1 + 3.97e31T + 7.64e62T^{2} \)
31 \( 1 + 2.09e32T + 1.34e64T^{2} \)
37 \( 1 - 2.50e33T + 2.70e67T^{2} \)
41 \( 1 + 4.85e34T + 2.23e69T^{2} \)
43 \( 1 + 1.71e35T + 1.73e70T^{2} \)
47 \( 1 + 5.66e35T + 7.94e71T^{2} \)
53 \( 1 + 3.68e36T + 1.39e74T^{2} \)
59 \( 1 + 2.83e37T + 1.40e76T^{2} \)
61 \( 1 - 1.88e38T + 5.87e76T^{2} \)
67 \( 1 - 4.53e37T + 3.32e78T^{2} \)
71 \( 1 - 9.40e39T + 4.01e79T^{2} \)
73 \( 1 - 1.85e40T + 1.32e80T^{2} \)
79 \( 1 - 2.01e40T + 3.96e81T^{2} \)
83 \( 1 - 5.04e40T + 3.31e82T^{2} \)
89 \( 1 - 2.49e41T + 6.66e83T^{2} \)
97 \( 1 - 5.50e42T + 2.69e85T^{2} \)
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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

−14.76106598841272696882443536687, −13.53244704435461337073574164602, −12.77781380583756211841277665729, −9.702647959702834978861160925650, −9.296499235348970975407832377435, −6.65277210365325693052747921469, −5.25975086481495194044518627305, −3.46990581254968739945337984940, −2.12413853973690171786007794616, 0, 2.12413853973690171786007794616, 3.46990581254968739945337984940, 5.25975086481495194044518627305, 6.65277210365325693052747921469, 9.296499235348970975407832377435, 9.702647959702834978861160925650, 12.77781380583756211841277665729, 13.53244704435461337073574164602, 14.76106598841272696882443536687

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