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  + 4.17·2-s + 2.46·3-s + 9.39·4-s − 7.54·5-s + 10.2·6-s + 4.58·7-s + 5.83·8-s − 20.9·9-s − 31.4·10-s + 26.9·11-s + 23.1·12-s − 15.6·13-s + 19.1·14-s − 18.6·15-s − 50.8·16-s + 27.2·17-s − 87.2·18-s + 38.3·19-s − 70.8·20-s + 11.3·21-s + 112.·22-s + 82.5·23-s + 14.3·24-s − 68.0·25-s − 65.2·26-s − 118.·27-s + 43.0·28-s + ⋯
L(s)  = 1  + 1.47·2-s + 0.474·3-s + 1.17·4-s − 0.674·5-s + 0.700·6-s + 0.247·7-s + 0.257·8-s − 0.774·9-s − 0.994·10-s + 0.737·11-s + 0.557·12-s − 0.333·13-s + 0.364·14-s − 0.320·15-s − 0.794·16-s + 0.388·17-s − 1.14·18-s + 0.462·19-s − 0.792·20-s + 0.117·21-s + 1.08·22-s + 0.748·23-s + 0.122·24-s − 0.544·25-s − 0.492·26-s − 0.842·27-s + 0.290·28-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  =  \((2,\ 43,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(2.62464\)
\(L(\frac12)\)  \(\approx\)  \(2.62464\)
\(L(\frac{5}{2})\)   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 \( 1 - 4.17T + 8T^{2} \)
3 \( 1 - 2.46T + 27T^{2} \)
5 \( 1 + 7.54T + 125T^{2} \)
7 \( 1 - 4.58T + 343T^{2} \)
11 \( 1 - 26.9T + 1.33e3T^{2} \)
13 \( 1 + 15.6T + 2.19e3T^{2} \)
17 \( 1 - 27.2T + 4.91e3T^{2} \)
19 \( 1 - 38.3T + 6.85e3T^{2} \)
23 \( 1 - 82.5T + 1.21e4T^{2} \)
29 \( 1 + 34.2T + 2.43e4T^{2} \)
31 \( 1 - 119.T + 2.97e4T^{2} \)
37 \( 1 - 378.T + 5.06e4T^{2} \)
41 \( 1 - 385.T + 6.89e4T^{2} \)
47 \( 1 - 271.T + 1.03e5T^{2} \)
53 \( 1 + 329.T + 1.48e5T^{2} \)
59 \( 1 + 173.T + 2.05e5T^{2} \)
61 \( 1 - 54.5T + 2.26e5T^{2} \)
67 \( 1 + 906.T + 3.00e5T^{2} \)
71 \( 1 + 621.T + 3.57e5T^{2} \)
73 \( 1 + 1.02e3T + 3.89e5T^{2} \)
79 \( 1 + 737.T + 4.93e5T^{2} \)
83 \( 1 - 558.T + 5.71e5T^{2} \)
89 \( 1 - 1.63e3T + 7.04e5T^{2} \)
97 \( 1 + 406.T + 9.12e5T^{2} \)
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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

−14.91669732999547393162830517398, −14.41953831857034778775594066943, −13.32456460779304134220588836705, −12.03275163129605300737676747088, −11.32006238996917064010693185014, −9.189227432129658339883458330683, −7.65373109568997716005663893938, −5.95291669955338180588365150927, −4.38867928280824437368629379094, −3.01978979060949517898100838695, 3.01978979060949517898100838695, 4.38867928280824437368629379094, 5.95291669955338180588365150927, 7.65373109568997716005663893938, 9.189227432129658339883458330683, 11.32006238996917064010693185014, 12.03275163129605300737676747088, 13.32456460779304134220588836705, 14.41953831857034778775594066943, 14.91669732999547393162830517398

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