Properties

Degree 2
Conductor 43
Sign $-0.769 + 0.639i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.90i·2-s − 11.0i·3-s + 16.3·4-s + 107. i·5-s − 76.4·6-s − 594. i·7-s − 554. i·8-s + 606.·9-s + 742.·10-s − 326.·11-s − 181. i·12-s − 3.32e3·13-s − 4.10e3·14-s + 1.19e3·15-s − 2.77e3·16-s + 1.73e3·17-s + ⋯
L(s)  = 1  − 0.862i·2-s − 0.410i·3-s + 0.255·4-s + 0.860i·5-s − 0.354·6-s − 1.73i·7-s − 1.08i·8-s + 0.831·9-s + 0.742·10-s − 0.245·11-s − 0.104i·12-s − 1.51·13-s − 1.49·14-s + 0.352·15-s − 0.678·16-s + 0.352·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.769 + 0.639i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.769 + 0.639i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.609772 - 1.68822i\)
\(L(\frac12)\)  \(\approx\)  \(0.609772 - 1.68822i\)
\(L(4)\)   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 \( 1 + (-6.11e4 + 5.08e4i)T \)
good2 \( 1 + 6.90iT - 64T^{2} \)
3 \( 1 + 11.0iT - 729T^{2} \)
5 \( 1 - 107. iT - 1.56e4T^{2} \)
7 \( 1 + 594. iT - 1.17e5T^{2} \)
11 \( 1 + 326.T + 1.77e6T^{2} \)
13 \( 1 + 3.32e3T + 4.82e6T^{2} \)
17 \( 1 - 1.73e3T + 2.41e7T^{2} \)
19 \( 1 + 6.99e3iT - 4.70e7T^{2} \)
23 \( 1 + 7.59e3T + 1.48e8T^{2} \)
29 \( 1 - 1.00e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.58e4T + 8.87e8T^{2} \)
37 \( 1 - 3.79e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.83e4T + 4.75e9T^{2} \)
47 \( 1 - 1.30e5T + 1.07e10T^{2} \)
53 \( 1 - 2.72e5T + 2.21e10T^{2} \)
59 \( 1 - 2.16e5T + 4.21e10T^{2} \)
61 \( 1 - 3.44e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.27e5T + 9.04e10T^{2} \)
71 \( 1 - 2.81e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.94e5iT - 1.51e11T^{2} \)
79 \( 1 + 4.97e5T + 2.43e11T^{2} \)
83 \( 1 + 1.43e5T + 3.26e11T^{2} \)
89 \( 1 - 5.23e5iT - 4.96e11T^{2} \)
97 \( 1 + 4.30e5T + 8.32e11T^{2} \)
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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

−13.96761902507883942650484124212, −12.95132747284415226769519076960, −11.76765492689962454337848632351, −10.41905509904692316856155381150, −10.09327134676063999862379853646, −7.26712829765033705539250479986, −7.01771737868492486808679724581, −4.20423708065706986589797669495, −2.60368732741431465554009586666, −0.855213602883476546646458016994, 2.24546500496106176198308224729, 4.88152901737494442264910615984, 5.86071756915218075604852564575, 7.59680484253913938550806498325, 8.794194196462902254947099353426, 10.00801402644735660856220822585, 11.94109368121151074937118152359, 12.55439266509926728154469170818, 14.51153727582252772451707735961, 15.37002200661033244566043649144

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