Properties

Degree 2
Conductor 43
Sign $-0.965 + 0.259i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.87i·2-s − 114. i·3-s + 221.·4-s − 623. i·5-s − 672.·6-s − 2.43e3i·7-s − 2.80e3i·8-s − 6.52e3·9-s − 3.66e3·10-s + 2.48e4·11-s − 2.53e4i·12-s + 8.70e3·13-s − 1.43e4·14-s − 7.13e4·15-s + 4.02e4·16-s + 1.01e4·17-s + ⋯
L(s)  = 1  − 0.367i·2-s − 1.41i·3-s + 0.865·4-s − 0.997i·5-s − 0.518·6-s − 1.01i·7-s − 0.684i·8-s − 0.994·9-s − 0.366·10-s + 1.69·11-s − 1.22i·12-s + 0.304·13-s − 0.372·14-s − 1.40·15-s + 0.613·16-s + 0.121·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.965 + 0.259i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.965 + 0.259i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.328906 - 2.49172i\)
\(L(\frac12)\)  \(\approx\)  \(0.328906 - 2.49172i\)
\(L(5)\)   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 \( 1 + (3.30e6 - 8.87e5i)T \)
good2 \( 1 + 5.87iT - 256T^{2} \)
3 \( 1 + 114. iT - 6.56e3T^{2} \)
5 \( 1 + 623. iT - 3.90e5T^{2} \)
7 \( 1 + 2.43e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.48e4T + 2.14e8T^{2} \)
13 \( 1 - 8.70e3T + 8.15e8T^{2} \)
17 \( 1 - 1.01e4T + 6.97e9T^{2} \)
19 \( 1 - 1.86e5iT - 1.69e10T^{2} \)
23 \( 1 + 3.87e5T + 7.83e10T^{2} \)
29 \( 1 - 1.03e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.46e5T + 8.52e11T^{2} \)
37 \( 1 - 2.77e6iT - 3.51e12T^{2} \)
41 \( 1 - 8.90e4T + 7.98e12T^{2} \)
47 \( 1 - 8.67e5T + 2.38e13T^{2} \)
53 \( 1 - 1.05e7T + 6.22e13T^{2} \)
59 \( 1 - 1.99e7T + 1.46e14T^{2} \)
61 \( 1 - 9.32e6iT - 1.91e14T^{2} \)
67 \( 1 + 1.83e7T + 4.06e14T^{2} \)
71 \( 1 + 8.73e6iT - 6.45e14T^{2} \)
73 \( 1 + 3.39e7iT - 8.06e14T^{2} \)
79 \( 1 - 2.74e7T + 1.51e15T^{2} \)
83 \( 1 + 6.16e7T + 2.25e15T^{2} \)
89 \( 1 + 9.46e6iT - 3.93e15T^{2} \)
97 \( 1 + 1.54e8T + 7.83e15T^{2} \)
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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

−13.41285181486887221899034808161, −12.27925267670217755582650935617, −11.79451656235787259295035898550, −10.15288939768821666071277237995, −8.390073378143182988985260794550, −7.15173129648073930957822884206, −6.20837768675777500843205357795, −3.82131811588116282389548212016, −1.54982406239232026318982965904, −1.09069196321194895350632066452, 2.44737678131276586809624688228, 3.89437542108192889005730631123, 5.75815312585002818748775113808, 6.85372973165241659269151208239, 8.758452190040863813088443249535, 9.934012552397659928931021558708, 11.17185330295453643004439427955, 11.83849658159973372705515853037, 14.23901559586935414316823590532, 15.06168142416800886475009611882

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