Properties

Degree 2
Conductor 43
Sign $-1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.4·2-s + 231.·3-s − 94.4·4-s − 2.58e3·5-s + 4.73e3·6-s − 1.76e3·7-s − 1.23e4·8-s + 3.41e4·9-s − 5.27e4·10-s − 7.45e4·11-s − 2.19e4·12-s + 7.10e3·13-s − 3.61e4·14-s − 5.99e5·15-s − 2.04e5·16-s − 2.68e5·17-s + 6.97e5·18-s + 5.72e5·19-s + 2.44e5·20-s − 4.10e5·21-s − 1.52e6·22-s + 1.28e6·23-s − 2.87e6·24-s + 4.72e6·25-s + 1.45e5·26-s + 3.35e6·27-s + 1.67e5·28-s + ⋯
L(s)  = 1  + 0.903·2-s + 1.65·3-s − 0.184·4-s − 1.84·5-s + 1.49·6-s − 0.278·7-s − 1.06·8-s + 1.73·9-s − 1.66·10-s − 1.53·11-s − 0.305·12-s + 0.0689·13-s − 0.251·14-s − 3.05·15-s − 0.781·16-s − 0.780·17-s + 1.56·18-s + 1.00·19-s + 0.341·20-s − 0.460·21-s − 1.38·22-s + 0.960·23-s − 1.76·24-s + 2.41·25-s + 0.0622·26-s + 1.21·27-s + 0.0514·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(9\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -1)\)
\(L(5)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{11}{2})\)   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 \( 1 + 3.41e6T \)
good2 \( 1 - 20.4T + 512T^{2} \)
3 \( 1 - 231.T + 1.96e4T^{2} \)
5 \( 1 + 2.58e3T + 1.95e6T^{2} \)
7 \( 1 + 1.76e3T + 4.03e7T^{2} \)
11 \( 1 + 7.45e4T + 2.35e9T^{2} \)
13 \( 1 - 7.10e3T + 1.06e10T^{2} \)
17 \( 1 + 2.68e5T + 1.18e11T^{2} \)
19 \( 1 - 5.72e5T + 3.22e11T^{2} \)
23 \( 1 - 1.28e6T + 1.80e12T^{2} \)
29 \( 1 + 6.39e6T + 1.45e13T^{2} \)
31 \( 1 + 2.00e6T + 2.64e13T^{2} \)
37 \( 1 - 1.49e7T + 1.29e14T^{2} \)
41 \( 1 + 1.77e7T + 3.27e14T^{2} \)
47 \( 1 - 1.13e7T + 1.11e15T^{2} \)
53 \( 1 + 1.02e8T + 3.29e15T^{2} \)
59 \( 1 + 8.07e7T + 8.66e15T^{2} \)
61 \( 1 - 6.21e7T + 1.16e16T^{2} \)
67 \( 1 - 1.55e8T + 2.72e16T^{2} \)
71 \( 1 + 1.87e8T + 4.58e16T^{2} \)
73 \( 1 - 4.82e7T + 5.88e16T^{2} \)
79 \( 1 + 4.54e8T + 1.19e17T^{2} \)
83 \( 1 - 1.00e8T + 1.86e17T^{2} \)
89 \( 1 - 1.13e9T + 3.50e17T^{2} \)
97 \( 1 + 2.37e8T + 7.60e17T^{2} \)
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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

−13.31446107433174582326732036476, −12.77458079436752988848735082196, −11.25042514737702690372388719159, −9.328370166068981420470219332001, −8.219610780124297346277444072784, −7.38666978151893807569355748029, −4.81540050246488019507660220790, −3.61671572173132762865312158428, −2.89510699653770513205567736260, 0, 2.89510699653770513205567736260, 3.61671572173132762865312158428, 4.81540050246488019507660220790, 7.38666978151893807569355748029, 8.219610780124297346277444072784, 9.328370166068981420470219332001, 11.25042514737702690372388719159, 12.77458079436752988848735082196, 13.31446107433174582326732036476

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