Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.48·2-s − 27.5·3-s − 25.8·4-s + 101.·5-s − 68.6·6-s + 15.7·7-s − 143.·8-s + 518.·9-s + 251.·10-s + 394.·11-s + 712.·12-s + 666.·13-s + 39.0·14-s − 2.79e3·15-s + 468.·16-s + 172.·17-s + 1.28e3·18-s − 1.28e3·19-s − 2.61e3·20-s − 433.·21-s + 981.·22-s + 569.·23-s + 3.96e3·24-s + 7.13e3·25-s + 1.65e3·26-s − 7.60e3·27-s − 405.·28-s + ⋯
L(s)  = 1  + 0.439·2-s − 1.77·3-s − 0.806·4-s + 1.81·5-s − 0.778·6-s + 0.121·7-s − 0.794·8-s + 2.13·9-s + 0.796·10-s + 0.983·11-s + 1.42·12-s + 1.09·13-s + 0.0532·14-s − 3.20·15-s + 0.457·16-s + 0.144·17-s + 0.938·18-s − 0.813·19-s − 1.46·20-s − 0.214·21-s + 0.432·22-s + 0.224·23-s + 1.40·24-s + 2.28·25-s + 0.481·26-s − 2.00·27-s − 0.0976·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.34739\)
\(L(\frac12)\)  \(\approx\)  \(1.34739\)
\(L(\frac{7}{2})\)   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 \( 1 - 1.84e3T \)
good2 \( 1 - 2.48T + 32T^{2} \)
3 \( 1 + 27.5T + 243T^{2} \)
5 \( 1 - 101.T + 3.12e3T^{2} \)
7 \( 1 - 15.7T + 1.68e4T^{2} \)
11 \( 1 - 394.T + 1.61e5T^{2} \)
13 \( 1 - 666.T + 3.71e5T^{2} \)
17 \( 1 - 172.T + 1.41e6T^{2} \)
19 \( 1 + 1.28e3T + 2.47e6T^{2} \)
23 \( 1 - 569.T + 6.43e6T^{2} \)
29 \( 1 + 6.32e3T + 2.05e7T^{2} \)
31 \( 1 - 7.79e3T + 2.86e7T^{2} \)
37 \( 1 - 1.62e4T + 6.93e7T^{2} \)
41 \( 1 - 7.45e3T + 1.15e8T^{2} \)
47 \( 1 + 5.62e3T + 2.29e8T^{2} \)
53 \( 1 - 2.24e4T + 4.18e8T^{2} \)
59 \( 1 + 9.06e3T + 7.14e8T^{2} \)
61 \( 1 + 1.82e4T + 8.44e8T^{2} \)
67 \( 1 + 2.74e4T + 1.35e9T^{2} \)
71 \( 1 + 1.28e4T + 1.80e9T^{2} \)
73 \( 1 + 6.34e4T + 2.07e9T^{2} \)
79 \( 1 + 1.91e3T + 3.07e9T^{2} \)
83 \( 1 + 5.21e4T + 3.93e9T^{2} \)
89 \( 1 - 6.61e4T + 5.58e9T^{2} \)
97 \( 1 - 3.74e4T + 8.58e9T^{2} \)
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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

−14.72160163966251253886466874473, −13.46518221297424372973490416558, −12.79458484229033959041639897619, −11.40765193792678328074662506792, −10.18760002839027116024419619809, −9.148691306199223975968332708479, −6.27770560505947578967942612289, −5.85045299373455112300848793120, −4.49823628971596812822810618290, −1.16736011070674858823024433785, 1.16736011070674858823024433785, 4.49823628971596812822810618290, 5.85045299373455112300848793120, 6.27770560505947578967942612289, 9.148691306199223975968332708479, 10.18760002839027116024419619809, 11.40765193792678328074662506792, 12.79458484229033959041639897619, 13.46518221297424372973490416558, 14.72160163966251253886466874473

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