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  − 3.15·2-s + 7.20·3-s + 1.96·4-s + 1.36·5-s − 22.7·6-s + 13.0·7-s + 19.0·8-s + 24.9·9-s − 4.30·10-s + 64.7·11-s + 14.1·12-s − 19.2·13-s − 41.0·14-s + 9.83·15-s − 75.8·16-s − 54.1·17-s − 78.8·18-s − 69.0·19-s + 2.67·20-s + 93.8·21-s − 204.·22-s + 29.6·23-s + 137.·24-s − 123.·25-s + 60.9·26-s − 14.6·27-s + 25.5·28-s + ⋯
L(s)  = 1  − 1.11·2-s + 1.38·3-s + 0.245·4-s + 0.121·5-s − 1.54·6-s + 0.702·7-s + 0.842·8-s + 0.924·9-s − 0.136·10-s + 1.77·11-s + 0.340·12-s − 0.411·13-s − 0.784·14-s + 0.169·15-s − 1.18·16-s − 0.772·17-s − 1.03·18-s − 0.833·19-s + 0.0299·20-s + 0.974·21-s − 1.98·22-s + 0.268·23-s + 1.16·24-s − 0.985·25-s + 0.459·26-s − 0.104·27-s + 0.172·28-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  =  \((2,\ 43,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.18786\)
\(L(\frac12)\)  \(\approx\)  \(1.18786\)
\(L(\frac{5}{2})\)   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 \( 1 + 3.15T + 8T^{2} \)
3 \( 1 - 7.20T + 27T^{2} \)
5 \( 1 - 1.36T + 125T^{2} \)
7 \( 1 - 13.0T + 343T^{2} \)
11 \( 1 - 64.7T + 1.33e3T^{2} \)
13 \( 1 + 19.2T + 2.19e3T^{2} \)
17 \( 1 + 54.1T + 4.91e3T^{2} \)
19 \( 1 + 69.0T + 6.85e3T^{2} \)
23 \( 1 - 29.6T + 1.21e4T^{2} \)
29 \( 1 - 13.1T + 2.43e4T^{2} \)
31 \( 1 - 185.T + 2.97e4T^{2} \)
37 \( 1 + 369.T + 5.06e4T^{2} \)
41 \( 1 + 294.T + 6.89e4T^{2} \)
47 \( 1 - 367.T + 1.03e5T^{2} \)
53 \( 1 - 708.T + 1.48e5T^{2} \)
59 \( 1 - 116.T + 2.05e5T^{2} \)
61 \( 1 - 218.T + 2.26e5T^{2} \)
67 \( 1 + 133.T + 3.00e5T^{2} \)
71 \( 1 + 926.T + 3.57e5T^{2} \)
73 \( 1 - 455.T + 3.89e5T^{2} \)
79 \( 1 + 620.T + 4.93e5T^{2} \)
83 \( 1 + 1.31e3T + 5.71e5T^{2} \)
89 \( 1 - 509.T + 7.04e5T^{2} \)
97 \( 1 - 965.T + 9.12e5T^{2} \)
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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

−15.32836647295734720501584343640, −14.31611260889005749018348234762, −13.54541070476797333550472357295, −11.68556770463828121257205546459, −10.11478032814635529065454491840, −8.957652713935566756127072289002, −8.446643352522337062228285424485, −7.04045787079637627980159620522, −4.18402275716880518852285964863, −1.83297946493770618690766357526, 1.83297946493770618690766357526, 4.18402275716880518852285964863, 7.04045787079637627980159620522, 8.446643352522337062228285424485, 8.957652713935566756127072289002, 10.11478032814635529065454491840, 11.68556770463828121257205546459, 13.54541070476797333550472357295, 14.31611260889005749018348234762, 15.32836647295734720501584343640

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