Properties

Degree 2
Conductor 43
Sign $-0.406 + 0.913i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 30.3i·2-s + 212. i·3-s + 105.·4-s + 1.63e3i·5-s − 6.45e3·6-s − 7.42e3i·7-s + 3.42e4i·8-s + 1.37e4·9-s − 4.94e4·10-s − 1.29e5·11-s + 2.24e4i·12-s − 4.77e5·13-s + 2.25e5·14-s − 3.47e5·15-s − 9.29e5·16-s − 2.39e5·17-s + ⋯
L(s)  = 1  + 0.947i·2-s + 0.876i·3-s + 0.102·4-s + 0.522i·5-s − 0.829·6-s − 0.441i·7-s + 1.04i·8-s + 0.232·9-s − 0.494·10-s − 0.805·11-s + 0.0900i·12-s − 1.28·13-s + 0.418·14-s − 0.457·15-s − 0.886·16-s − 0.168·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.406 + 0.913i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.406 + 0.913i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.618051 - 0.951193i\)
\(L(\frac12)\)  \(\approx\)  \(0.618051 - 0.951193i\)
\(L(6)\)   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 \( 1 + (-5.97e7 + 1.34e8i)T \)
good2 \( 1 - 30.3iT - 1.02e3T^{2} \)
3 \( 1 - 212. iT - 5.90e4T^{2} \)
5 \( 1 - 1.63e3iT - 9.76e6T^{2} \)
7 \( 1 + 7.42e3iT - 2.82e8T^{2} \)
11 \( 1 + 1.29e5T + 2.59e10T^{2} \)
13 \( 1 + 4.77e5T + 1.37e11T^{2} \)
17 \( 1 + 2.39e5T + 2.01e12T^{2} \)
19 \( 1 - 2.40e6iT - 6.13e12T^{2} \)
23 \( 1 + 1.07e7T + 4.14e13T^{2} \)
29 \( 1 + 3.05e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.70e7T + 8.19e14T^{2} \)
37 \( 1 + 2.92e7iT - 4.80e15T^{2} \)
41 \( 1 + 6.33e7T + 1.34e16T^{2} \)
47 \( 1 - 1.73e7T + 5.25e16T^{2} \)
53 \( 1 + 9.26e7T + 1.74e17T^{2} \)
59 \( 1 + 8.79e8T + 5.11e17T^{2} \)
61 \( 1 - 1.35e9iT - 7.13e17T^{2} \)
67 \( 1 - 1.85e8T + 1.82e18T^{2} \)
71 \( 1 - 3.17e9iT - 3.25e18T^{2} \)
73 \( 1 - 2.67e9iT - 4.29e18T^{2} \)
79 \( 1 - 5.12e9T + 9.46e18T^{2} \)
83 \( 1 + 9.86e8T + 1.55e19T^{2} \)
89 \( 1 + 5.62e8iT - 3.11e19T^{2} \)
97 \( 1 + 1.18e10T + 7.37e19T^{2} \)
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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

−14.87153427586088526902501960468, −13.88486604985476951374611476868, −12.11727737667503141375899558984, −10.62579988747163589101341989771, −9.893495292470529137598865106832, −8.026864133352469682169227254411, −7.06211197029164993091564070736, −5.61137557331428127963407189711, −4.25082777761779501982091669728, −2.41465199832715270326488687033, 0.33414145984390626745334166683, 1.73099140754921972797733052857, 2.75249427574775830259825378147, 4.79014743511544297129830138604, 6.63416016844782745109634967300, 7.82691245926387876173416733327, 9.474909641900431455802763150281, 10.66921879555606778599625475896, 12.16488535718634817095411979117, 12.50064700194499239461911707944

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