Properties

Degree 2
Conductor 43
Sign $-0.296 - 0.954i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.2i·2-s + (14.4 − 8.32i)3-s − 84.9·4-s + (131. − 75.7i)5-s + (101. + 176. i)6-s + (221. + 128. i)7-s − 255. i·8-s + (−225. + 391. i)9-s + (924. + 1.60e3i)10-s + 1.82e3·11-s + (−1.22e3 + 707. i)12-s + (−765. + 1.32e3i)13-s + (−1.56e3 + 2.70e3i)14-s + (1.26e3 − 2.18e3i)15-s − 2.31e3·16-s + (3.55e3 − 6.15e3i)17-s + ⋯
L(s)  = 1  + 1.52i·2-s + (0.534 − 0.308i)3-s − 1.32·4-s + (1.04 − 0.606i)5-s + (0.470 + 0.814i)6-s + (0.646 + 0.373i)7-s − 0.499i·8-s + (−0.309 + 0.536i)9-s + (0.924 + 1.60i)10-s + 1.37·11-s + (−0.709 + 0.409i)12-s + (−0.348 + 0.603i)13-s + (−0.569 + 0.986i)14-s + (0.373 − 0.647i)15-s − 0.565·16-s + (0.722 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.296 - 0.954i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.296 - 0.954i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.44883 + 1.96749i\)
\(L(\frac12)\)  \(\approx\)  \(1.44883 + 1.96749i\)
\(L(4)\)   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 \( 1 + (-7.76e4 + 1.69e4i)T \)
good2 \( 1 - 12.2iT - 64T^{2} \)
3 \( 1 + (-14.4 + 8.32i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-131. + 75.7i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-221. - 128. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 - 1.82e3T + 1.77e6T^{2} \)
13 \( 1 + (765. - 1.32e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-3.55e3 + 6.15e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (6.45e3 - 3.72e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-3.15e3 - 5.45e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-9.26e3 - 5.35e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (9.13e3 + 1.58e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-6.48e4 + 3.74e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 8.95e4T + 4.75e9T^{2} \)
47 \( 1 + 5.62e4T + 1.07e10T^{2} \)
53 \( 1 + (2.12e4 + 3.68e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 - 2.44e5T + 4.21e10T^{2} \)
61 \( 1 + (1.25e5 + 7.25e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (9.67e4 + 1.67e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (4.21e5 + 2.43e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (5.07e5 + 2.92e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (2.02e4 - 3.50e4i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (1.93e5 + 3.35e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (-6.79e5 + 3.92e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 6.03e4T + 8.32e11T^{2} \)
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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

−14.68805184742040940566475845723, −14.27969789616123431834743090877, −13.32549088297336350946415344885, −11.66394063725851910616873250234, −9.411209130699088475910987054625, −8.655299250947249054181407005409, −7.39757893117699817488262776863, −5.99288791687153772732792989008, −4.86528019489267441208563786748, −1.87613629111261097386912485073, 1.33323539579376744681146282779, 2.77713691638316501224775957426, 4.12882326841314822188818717742, 6.37588285972600540736458746081, 8.660990980730453323491483300315, 9.813922885928946840825630917165, 10.59743322042576835351276334596, 11.79915982492060724512484317986, 13.03335718654324133704069856515, 14.31103731221001837934451440115

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