Properties

Degree 2
Conductor 43
Sign $0.623 - 0.781i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.1i·2-s − 22.6i·3-s − 108.·4-s − 138. i·5-s + 297.·6-s + 295. i·7-s − 584. i·8-s + 216.·9-s + 1.82e3·10-s + 1.63e3·11-s + 2.45e3i·12-s + 3.92e3·13-s − 3.88e3·14-s − 3.14e3·15-s + 732.·16-s − 3.26e3·17-s + ⋯
L(s)  = 1  + 1.64i·2-s − 0.838i·3-s − 1.69·4-s − 1.11i·5-s + 1.37·6-s + 0.861i·7-s − 1.14i·8-s + 0.297·9-s + 1.82·10-s + 1.23·11-s + 1.42i·12-s + 1.78·13-s − 1.41·14-s − 0.931·15-s + 0.178·16-s − 0.664·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.623 - 0.781i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.623 - 0.781i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.61926 + 0.779760i\)
\(L(\frac12)\)  \(\approx\)  \(1.61926 + 0.779760i\)
\(L(4)\)   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 \( 1 + (4.95e4 - 6.21e4i)T \)
good2 \( 1 - 13.1iT - 64T^{2} \)
3 \( 1 + 22.6iT - 729T^{2} \)
5 \( 1 + 138. iT - 1.56e4T^{2} \)
7 \( 1 - 295. iT - 1.17e5T^{2} \)
11 \( 1 - 1.63e3T + 1.77e6T^{2} \)
13 \( 1 - 3.92e3T + 4.82e6T^{2} \)
17 \( 1 + 3.26e3T + 2.41e7T^{2} \)
19 \( 1 + 8.53e3iT - 4.70e7T^{2} \)
23 \( 1 - 9.92e3T + 1.48e8T^{2} \)
29 \( 1 - 1.05e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.88e4T + 8.87e8T^{2} \)
37 \( 1 - 9.01e3iT - 2.56e9T^{2} \)
41 \( 1 + 8.39e4T + 4.75e9T^{2} \)
47 \( 1 - 9.89e4T + 1.07e10T^{2} \)
53 \( 1 - 4.94e4T + 2.21e10T^{2} \)
59 \( 1 + 3.29e4T + 4.21e10T^{2} \)
61 \( 1 - 2.94e5iT - 5.15e10T^{2} \)
67 \( 1 + 3.52e5T + 9.04e10T^{2} \)
71 \( 1 + 4.12e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.80e5iT - 1.51e11T^{2} \)
79 \( 1 + 7.52e5T + 2.43e11T^{2} \)
83 \( 1 + 1.64e5T + 3.26e11T^{2} \)
89 \( 1 + 9.04e5iT - 4.96e11T^{2} \)
97 \( 1 - 9.77e5T + 8.32e11T^{2} \)
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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

−15.19728901811364578540808107892, −13.64368577575997538753658144192, −13.02757460641973157133616121836, −11.65902409474664380918323444113, −8.886092064440888658068660289311, −8.679126837206210031581417269108, −6.96707973457963395162241971481, −6.05834628726457194363731279856, −4.58657561990568216652621687049, −1.14928651980621580062048321102, 1.35145926537174457466475865199, 3.47974930001961944909266377375, 4.09501423572819286706062510059, 6.68229196316470812878998931154, 8.940316124400253997730369536804, 10.23823738487733178941211780106, 10.73837203165939763299925575353, 11.67587483074239807155492525347, 13.28838440288528556398554042848, 14.20431480555706132275474394054

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