Properties

Degree 2
Conductor 43
Sign $0.525 + 0.850i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 24.2i·2-s − 13.8i·3-s − 333.·4-s + 427. i·5-s − 335.·6-s + 2.56e3i·7-s + 1.89e3i·8-s + 6.37e3·9-s + 1.03e4·10-s + 3.29e3·11-s + 4.61e3i·12-s + 1.78e3·13-s + 6.22e4·14-s + 5.90e3·15-s − 3.94e4·16-s + 1.61e5·17-s + ⋯
L(s)  = 1  − 1.51i·2-s − 0.170i·3-s − 1.30·4-s + 0.684i·5-s − 0.258·6-s + 1.06i·7-s + 0.462i·8-s + 0.970·9-s + 1.03·10-s + 0.225·11-s + 0.222i·12-s + 0.0624·13-s + 1.62·14-s + 0.116·15-s − 0.602·16-s + 1.93·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.525 + 0.850i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.525 + 0.850i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.69070 - 0.942438i\)
\(L(\frac12)\)  \(\approx\)  \(1.69070 - 0.942438i\)
\(L(5)\)   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 \( 1 + (-1.79e6 - 2.90e6i)T \)
good2 \( 1 + 24.2iT - 256T^{2} \)
3 \( 1 + 13.8iT - 6.56e3T^{2} \)
5 \( 1 - 427. iT - 3.90e5T^{2} \)
7 \( 1 - 2.56e3iT - 5.76e6T^{2} \)
11 \( 1 - 3.29e3T + 2.14e8T^{2} \)
13 \( 1 - 1.78e3T + 8.15e8T^{2} \)
17 \( 1 - 1.61e5T + 6.97e9T^{2} \)
19 \( 1 - 1.88e5iT - 1.69e10T^{2} \)
23 \( 1 + 4.30e5T + 7.83e10T^{2} \)
29 \( 1 + 1.23e6iT - 5.00e11T^{2} \)
31 \( 1 - 8.19e5T + 8.52e11T^{2} \)
37 \( 1 - 1.89e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.24e6T + 7.98e12T^{2} \)
47 \( 1 - 3.12e6T + 2.38e13T^{2} \)
53 \( 1 - 6.47e6T + 6.22e13T^{2} \)
59 \( 1 + 1.77e5T + 1.46e14T^{2} \)
61 \( 1 - 1.30e7iT - 1.91e14T^{2} \)
67 \( 1 + 1.94e7T + 4.06e14T^{2} \)
71 \( 1 - 1.97e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.15e7iT - 8.06e14T^{2} \)
79 \( 1 - 8.29e6T + 1.51e15T^{2} \)
83 \( 1 - 4.28e7T + 2.25e15T^{2} \)
89 \( 1 - 2.70e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.00e8T + 7.83e15T^{2} \)
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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

−13.71526302722188978516330805189, −12.20721162037831967359768656648, −11.99793160795639495492102048110, −10.31314802488710758018209530143, −9.734338912506677957629054323442, −7.929187995691876013428163884333, −6.04344095820293799568549495437, −3.96870414845807537169645004866, −2.61167721560011482251233663844, −1.30201666516492542882992296176, 0.906506023504202229375683455935, 4.11876159361576966072406739611, 5.30043444646400366824245508457, 6.90210346680321964372905150360, 7.76937889854151367362420861387, 9.151658496384113886271785713895, 10.46205592225736311787421315362, 12.38884845289283378322800872506, 13.64655828446114191675650024637, 14.51274721036125862051821869266

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