Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.15·2-s − 6.49·3-s + 18.5·4-s + 17.2·5-s − 33.4·6-s − 23.3·7-s + 54.3·8-s + 15.2·9-s + 88.9·10-s − 60.5·11-s − 120.·12-s + 10.9·13-s − 120.·14-s − 112.·15-s + 131.·16-s − 3.57·17-s + 78.5·18-s + 33.2·19-s + 320.·20-s + 151.·21-s − 312.·22-s + 63.7·23-s − 353.·24-s + 173.·25-s + 56.3·26-s + 76.4·27-s − 432.·28-s + ⋯
L(s)  = 1  + 1.82·2-s − 1.25·3-s + 2.31·4-s + 1.54·5-s − 2.27·6-s − 1.25·7-s + 2.40·8-s + 0.564·9-s + 2.81·10-s − 1.65·11-s − 2.90·12-s + 0.233·13-s − 2.29·14-s − 1.93·15-s + 2.05·16-s − 0.0509·17-s + 1.02·18-s + 0.401·19-s + 3.58·20-s + 1.57·21-s − 3.02·22-s + 0.577·23-s − 3.00·24-s + 1.38·25-s + 0.425·26-s + 0.544·27-s − 2.91·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(2.56032\)
\(L(\frac12)\)  \(\approx\)  \(2.56032\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + 43T \)
good2 \( 1 - 5.15T + 8T^{2} \)
3 \( 1 + 6.49T + 27T^{2} \)
5 \( 1 - 17.2T + 125T^{2} \)
7 \( 1 + 23.3T + 343T^{2} \)
11 \( 1 + 60.5T + 1.33e3T^{2} \)
13 \( 1 - 10.9T + 2.19e3T^{2} \)
17 \( 1 + 3.57T + 4.91e3T^{2} \)
19 \( 1 - 33.2T + 6.85e3T^{2} \)
23 \( 1 - 63.7T + 1.21e4T^{2} \)
29 \( 1 + 89.3T + 2.43e4T^{2} \)
31 \( 1 - 222.T + 2.97e4T^{2} \)
37 \( 1 + 59.6T + 5.06e4T^{2} \)
41 \( 1 + 143.T + 6.89e4T^{2} \)
47 \( 1 - 379.T + 1.03e5T^{2} \)
53 \( 1 + 150.T + 1.48e5T^{2} \)
59 \( 1 - 207.T + 2.05e5T^{2} \)
61 \( 1 + 486.T + 2.26e5T^{2} \)
67 \( 1 - 1.01e3T + 3.00e5T^{2} \)
71 \( 1 - 13.8T + 3.57e5T^{2} \)
73 \( 1 - 411.T + 3.89e5T^{2} \)
79 \( 1 + 1.31e3T + 4.93e5T^{2} \)
83 \( 1 - 813.T + 5.71e5T^{2} \)
89 \( 1 + 350.T + 7.04e5T^{2} \)
97 \( 1 + 1.18e3T + 9.12e5T^{2} \)
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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

−15.48801412621755453619771197275, −13.80727724237216374240032603737, −13.15987147008112891119161368636, −12.40205058679316898501127675983, −10.93586598781839085607401078505, −10.00896459683599676926309892380, −6.73845211370475354777199837234, −5.83976303890908103743692306354, −5.15330130451801237043567001255, −2.77663447143581097021597185479, 2.77663447143581097021597185479, 5.15330130451801237043567001255, 5.83976303890908103743692306354, 6.73845211370475354777199837234, 10.00896459683599676926309892380, 10.93586598781839085607401078505, 12.40205058679316898501127675983, 13.15987147008112891119161368636, 13.80727724237216374240032603737, 15.48801412621755453619771197275

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