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  − 4.31·2-s − 23.8·3-s − 13.3·4-s − 52.5·5-s + 102.·6-s − 174.·7-s + 195.·8-s + 325.·9-s + 226.·10-s − 447.·11-s + 319.·12-s + 669.·13-s + 754.·14-s + 1.25e3·15-s − 416.·16-s − 849.·17-s − 1.40e3·18-s − 1.28e3·19-s + 703.·20-s + 4.16e3·21-s + 1.93e3·22-s − 378.·23-s − 4.66e3·24-s − 359.·25-s − 2.88e3·26-s − 1.97e3·27-s + 2.33e3·28-s + ⋯
L(s)  = 1  − 0.762·2-s − 1.52·3-s − 0.418·4-s − 0.940·5-s + 1.16·6-s − 1.34·7-s + 1.08·8-s + 1.34·9-s + 0.717·10-s − 1.11·11-s + 0.639·12-s + 1.09·13-s + 1.02·14-s + 1.43·15-s − 0.407·16-s − 0.713·17-s − 1.02·18-s − 0.819·19-s + 0.393·20-s + 2.06·21-s + 0.851·22-s − 0.149·23-s − 1.65·24-s − 0.115·25-s − 0.837·26-s − 0.520·27-s + 0.563·28-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  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(0.126326\)
\(L(\frac12)\)  \(\approx\)  \(0.126326\)
\(L(\frac{7}{2})\)   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 \( 1 + 4.31T + 32T^{2} \)
3 \( 1 + 23.8T + 243T^{2} \)
5 \( 1 + 52.5T + 3.12e3T^{2} \)
7 \( 1 + 174.T + 1.68e4T^{2} \)
11 \( 1 + 447.T + 1.61e5T^{2} \)
13 \( 1 - 669.T + 3.71e5T^{2} \)
17 \( 1 + 849.T + 1.41e6T^{2} \)
19 \( 1 + 1.28e3T + 2.47e6T^{2} \)
23 \( 1 + 378.T + 6.43e6T^{2} \)
29 \( 1 - 765.T + 2.05e7T^{2} \)
31 \( 1 + 7.09e3T + 2.86e7T^{2} \)
37 \( 1 + 7.90e3T + 6.93e7T^{2} \)
41 \( 1 - 1.28e4T + 1.15e8T^{2} \)
47 \( 1 - 2.67e4T + 2.29e8T^{2} \)
53 \( 1 - 3.00e4T + 4.18e8T^{2} \)
59 \( 1 - 1.24e3T + 7.14e8T^{2} \)
61 \( 1 + 4.84e4T + 8.44e8T^{2} \)
67 \( 1 - 6.70e4T + 1.35e9T^{2} \)
71 \( 1 + 7.45e4T + 1.80e9T^{2} \)
73 \( 1 + 1.80e4T + 2.07e9T^{2} \)
79 \( 1 + 6.32e4T + 3.07e9T^{2} \)
83 \( 1 + 8.80e4T + 3.93e9T^{2} \)
89 \( 1 - 5.03e4T + 5.58e9T^{2} \)
97 \( 1 - 1.75e4T + 8.58e9T^{2} \)
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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

−15.73961968138714541544473240701, −13.32410699933760574850095596762, −12.50478319287197356409885994240, −11.03195992116345356465867149024, −10.34764918951052293666627721665, −8.797880452020365467018047324036, −7.25022158904520124227393199304, −5.81095157231506827861527316982, −4.11832708096723618849682078976, −0.35875812059184369204644146503, 0.35875812059184369204644146503, 4.11832708096723618849682078976, 5.81095157231506827861527316982, 7.25022158904520124227393199304, 8.797880452020365467018047324036, 10.34764918951052293666627721665, 11.03195992116345356465867149024, 12.50478319287197356409885994240, 13.32410699933760574850095596762, 15.73961968138714541544473240701

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