Properties

Degree 2
Conductor 43
Sign $-0.994 + 0.104i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.14i·2-s − 19.7i·3-s + 13.0·4-s − 151. i·5-s − 141.·6-s + 62.8i·7-s − 549. i·8-s + 338.·9-s − 1.08e3·10-s − 497.·11-s − 256. i·12-s + 850.·13-s + 449.·14-s − 2.99e3·15-s − 3.09e3·16-s − 8.04e3·17-s + ⋯
L(s)  = 1  − 0.892i·2-s − 0.731i·3-s + 0.203·4-s − 1.21i·5-s − 0.653·6-s + 0.183i·7-s − 1.07i·8-s + 0.464·9-s − 1.08·10-s − 0.374·11-s − 0.148i·12-s + 0.387·13-s + 0.163·14-s − 0.888·15-s − 0.755·16-s − 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.994 + 0.104i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.994 + 0.104i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.0962520 - 1.83380i\)
\(L(\frac12)\)  \(\approx\)  \(0.0962520 - 1.83380i\)
\(L(4)\)   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 \( 1 + (-7.90e4 + 8.32e3i)T \)
good2 \( 1 + 7.14iT - 64T^{2} \)
3 \( 1 + 19.7iT - 729T^{2} \)
5 \( 1 + 151. iT - 1.56e4T^{2} \)
7 \( 1 - 62.8iT - 1.17e5T^{2} \)
11 \( 1 + 497.T + 1.77e6T^{2} \)
13 \( 1 - 850.T + 4.82e6T^{2} \)
17 \( 1 + 8.04e3T + 2.41e7T^{2} \)
19 \( 1 - 1.14e4iT - 4.70e7T^{2} \)
23 \( 1 - 6.20e3T + 1.48e8T^{2} \)
29 \( 1 + 2.38e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.69e4T + 8.87e8T^{2} \)
37 \( 1 + 2.35e4iT - 2.56e9T^{2} \)
41 \( 1 - 8.73e4T + 4.75e9T^{2} \)
47 \( 1 + 3.90e4T + 1.07e10T^{2} \)
53 \( 1 + 1.78e5T + 2.21e10T^{2} \)
59 \( 1 - 1.99e4T + 4.21e10T^{2} \)
61 \( 1 - 3.49e4iT - 5.15e10T^{2} \)
67 \( 1 + 2.99e5T + 9.04e10T^{2} \)
71 \( 1 + 3.62e5iT - 1.28e11T^{2} \)
73 \( 1 + 8.99e4iT - 1.51e11T^{2} \)
79 \( 1 - 4.34e5T + 2.43e11T^{2} \)
83 \( 1 - 6.62e5T + 3.26e11T^{2} \)
89 \( 1 - 1.26e6iT - 4.96e11T^{2} \)
97 \( 1 - 8.37e4T + 8.32e11T^{2} \)
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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

−13.48864923401415226960463890594, −12.73428638032138640550665587394, −12.02094557248216795975431464162, −10.67280640936547227298607707513, −9.272856670217719743812983828089, −7.84622381068558516038449900007, −6.26627022539589924686746471782, −4.27314024836927261905783803841, −2.11546369424120256196312714218, −0.891080355346916832721866019758, 2.72324676982548523472340457546, 4.69367719737750468804524896277, 6.50296527277967836271175503440, 7.27982759677832434740628690002, 8.998658649808456453313428265277, 10.68372292665398635756380668428, 11.14528217971961430205670113455, 13.28849916790027669728186299900, 14.55101801311557186527256432811, 15.51824084973544794534445191038

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