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  − 8.86·2-s + 1.50·3-s + 46.5·4-s − 37.8·5-s − 13.3·6-s − 124.·7-s − 129.·8-s − 240.·9-s + 335.·10-s + 590.·11-s + 69.9·12-s + 434.·13-s + 1.10e3·14-s − 56.8·15-s − 343.·16-s + 1.92e3·17-s + 2.13e3·18-s + 654.·19-s − 1.76e3·20-s − 187.·21-s − 5.23e3·22-s + 2.80e3·23-s − 194.·24-s − 1.69e3·25-s − 3.85e3·26-s − 726.·27-s − 5.81e3·28-s + ⋯
L(s)  = 1  − 1.56·2-s + 0.0963·3-s + 1.45·4-s − 0.676·5-s − 0.150·6-s − 0.962·7-s − 0.714·8-s − 0.990·9-s + 1.06·10-s + 1.47·11-s + 0.140·12-s + 0.713·13-s + 1.50·14-s − 0.0651·15-s − 0.335·16-s + 1.61·17-s + 1.55·18-s + 0.415·19-s − 0.985·20-s − 0.0926·21-s − 2.30·22-s + 1.10·23-s − 0.0688·24-s − 0.542·25-s − 1.11·26-s − 0.191·27-s − 1.40·28-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  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(0.605902\)
\(L(\frac12)\)  \(\approx\)  \(0.605902\)
\(L(\frac{7}{2})\)   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 \( 1 + 8.86T + 32T^{2} \)
3 \( 1 - 1.50T + 243T^{2} \)
5 \( 1 + 37.8T + 3.12e3T^{2} \)
7 \( 1 + 124.T + 1.68e4T^{2} \)
11 \( 1 - 590.T + 1.61e5T^{2} \)
13 \( 1 - 434.T + 3.71e5T^{2} \)
17 \( 1 - 1.92e3T + 1.41e6T^{2} \)
19 \( 1 - 654.T + 2.47e6T^{2} \)
23 \( 1 - 2.80e3T + 6.43e6T^{2} \)
29 \( 1 + 1.45e3T + 2.05e7T^{2} \)
31 \( 1 - 4.41e3T + 2.86e7T^{2} \)
37 \( 1 - 3.75e3T + 6.93e7T^{2} \)
41 \( 1 - 1.97e3T + 1.15e8T^{2} \)
47 \( 1 - 2.20e3T + 2.29e8T^{2} \)
53 \( 1 - 2.49e4T + 4.18e8T^{2} \)
59 \( 1 + 4.27e4T + 7.14e8T^{2} \)
61 \( 1 + 2.10e4T + 8.44e8T^{2} \)
67 \( 1 + 2.52e4T + 1.35e9T^{2} \)
71 \( 1 - 4.80e4T + 1.80e9T^{2} \)
73 \( 1 - 5.88e4T + 2.07e9T^{2} \)
79 \( 1 - 9.27e4T + 3.07e9T^{2} \)
83 \( 1 + 1.84e3T + 3.93e9T^{2} \)
89 \( 1 + 7.03e4T + 5.58e9T^{2} \)
97 \( 1 + 8.88e4T + 8.58e9T^{2} \)
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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

−15.34188352102271567701016646120, −13.93458291464995425414501630511, −12.08317128609614820257836666535, −11.16523757763141079317222780046, −9.711032950402186267292487581400, −8.856126538390184116956701603357, −7.66817218950546579436774081648, −6.27922955985001549821956599686, −3.39043826316152718059482802822, −0.872117925969098084697913757559, 0.872117925969098084697913757559, 3.39043826316152718059482802822, 6.27922955985001549821956599686, 7.66817218950546579436774081648, 8.856126538390184116956701603357, 9.711032950402186267292487581400, 11.16523757763141079317222780046, 12.08317128609614820257836666535, 13.93458291464995425414501630511, 15.34188352102271567701016646120

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