Properties

Degree 2
Conductor 43
Sign $0.599 + 0.800i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.9i·2-s + 271. i·3-s + 127.·4-s − 203. i·5-s + 8.14e3·6-s − 3.47e3i·7-s − 3.44e4i·8-s − 1.49e4·9-s − 6.09e3·10-s − 1.86e5·11-s + 3.46e4i·12-s + 1.80e5·13-s − 1.04e5·14-s + 5.53e4·15-s − 9.02e5·16-s + 2.31e6·17-s + ⋯
L(s)  = 1  − 0.935i·2-s + 1.11i·3-s + 0.124·4-s − 0.0651i·5-s + 1.04·6-s − 0.207i·7-s − 1.05i·8-s − 0.252·9-s − 0.0609·10-s − 1.15·11-s + 0.139i·12-s + 0.487·13-s − 0.193·14-s + 0.0728·15-s − 0.860·16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.599 + 0.800i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.599 + 0.800i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(2.05065 - 1.02548i\)
\(L(\frac12)\)  \(\approx\)  \(2.05065 - 1.02548i\)
\(L(6)\)   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 \( 1 + (8.81e7 + 1.17e8i)T \)
good2 \( 1 + 29.9iT - 1.02e3T^{2} \)
3 \( 1 - 271. iT - 5.90e4T^{2} \)
5 \( 1 + 203. iT - 9.76e6T^{2} \)
7 \( 1 + 3.47e3iT - 2.82e8T^{2} \)
11 \( 1 + 1.86e5T + 2.59e10T^{2} \)
13 \( 1 - 1.80e5T + 1.37e11T^{2} \)
17 \( 1 - 2.31e6T + 2.01e12T^{2} \)
19 \( 1 + 1.81e6iT - 6.13e12T^{2} \)
23 \( 1 - 3.25e6T + 4.14e13T^{2} \)
29 \( 1 - 1.93e6iT - 4.20e14T^{2} \)
31 \( 1 - 1.24e7T + 8.19e14T^{2} \)
37 \( 1 + 1.29e8iT - 4.80e15T^{2} \)
41 \( 1 - 5.75e7T + 1.34e16T^{2} \)
47 \( 1 - 2.19e8T + 5.25e16T^{2} \)
53 \( 1 - 2.24e8T + 1.74e17T^{2} \)
59 \( 1 - 4.63e8T + 5.11e17T^{2} \)
61 \( 1 - 1.32e9iT - 7.13e17T^{2} \)
67 \( 1 + 9.00e8T + 1.82e18T^{2} \)
71 \( 1 + 5.54e8iT - 3.25e18T^{2} \)
73 \( 1 + 2.21e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.15e9T + 9.46e18T^{2} \)
83 \( 1 - 1.02e9T + 1.55e19T^{2} \)
89 \( 1 - 6.93e9iT - 3.11e19T^{2} \)
97 \( 1 - 9.01e9T + 7.37e19T^{2} \)
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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

−13.31541901029068327254434884744, −12.21181133190189801544215935958, −10.76227560182179946211255901349, −10.37993041705623078036590814381, −9.137512646729730825307726768617, −7.32871670879153508973153792182, −5.35075747899649251885592707954, −3.89316894843314068045512845529, −2.74291795987960040421180289530, −0.876609248419893901335776567495, 1.21772605194688622356920837890, 2.75635742437829086752992521674, 5.32489504072585277824042019535, 6.43710574854777888878885373003, 7.57348101900690113523913548102, 8.280832307745905994414981995158, 10.33102599600282958354983667856, 11.81495252990652799097578223539, 12.86977946267408756831973362017, 13.99788244835015261787737926615

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