Properties

Degree 2
Conductor 43
Sign $0.969 + 0.243i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.567i·2-s + 31.6i·3-s + 63.6·4-s − 195. i·5-s − 17.9·6-s − 418. i·7-s + 72.4i·8-s − 272.·9-s + 110.·10-s + 1.18e3·11-s + 2.01e3i·12-s − 15.3·13-s + 237.·14-s + 6.18e3·15-s + 4.03e3·16-s − 2.66e3·17-s + ⋯
L(s)  = 1  + 0.0709i·2-s + 1.17i·3-s + 0.994·4-s − 1.56i·5-s − 0.0831·6-s − 1.21i·7-s + 0.141i·8-s − 0.373·9-s + 0.110·10-s + 0.890·11-s + 1.16i·12-s − 0.00700·13-s + 0.0865·14-s + 1.83·15-s + 0.984·16-s − 0.541·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.969 + 0.243i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.969 + 0.243i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(2.18532 - 0.270512i\)
\(L(\frac12)\)  \(\approx\)  \(2.18532 - 0.270512i\)
\(L(4)\)   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 \( 1 + (7.71e4 + 1.93e4i)T \)
good2 \( 1 - 0.567iT - 64T^{2} \)
3 \( 1 - 31.6iT - 729T^{2} \)
5 \( 1 + 195. iT - 1.56e4T^{2} \)
7 \( 1 + 418. iT - 1.17e5T^{2} \)
11 \( 1 - 1.18e3T + 1.77e6T^{2} \)
13 \( 1 + 15.3T + 4.82e6T^{2} \)
17 \( 1 + 2.66e3T + 2.41e7T^{2} \)
19 \( 1 + 2.35e3iT - 4.70e7T^{2} \)
23 \( 1 - 2.89e3T + 1.48e8T^{2} \)
29 \( 1 - 2.77e4iT - 5.94e8T^{2} \)
31 \( 1 - 5.09e4T + 8.87e8T^{2} \)
37 \( 1 + 6.42e4iT - 2.56e9T^{2} \)
41 \( 1 + 8.75e4T + 4.75e9T^{2} \)
47 \( 1 + 1.11e5T + 1.07e10T^{2} \)
53 \( 1 - 2.21e5T + 2.21e10T^{2} \)
59 \( 1 + 1.96e5T + 4.21e10T^{2} \)
61 \( 1 - 1.91e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.80e5T + 9.04e10T^{2} \)
71 \( 1 + 1.67e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.52e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.21e5T + 2.43e11T^{2} \)
83 \( 1 - 4.93e4T + 3.26e11T^{2} \)
89 \( 1 - 1.33e6iT - 4.96e11T^{2} \)
97 \( 1 - 8.30e5T + 8.32e11T^{2} \)
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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

−14.97851647862693656523706484888, −13.49561784973308482411647630061, −12.14188394558810659007063832013, −10.91246717271447016920236764819, −9.821607255748121757244907946345, −8.584834017780158328796413617742, −6.89425675115092453526579329343, −4.99079348134402525991065050589, −3.87159875827842203685844737005, −1.17276957545697962098582879771, 1.86074671692162687021041730702, 2.93275482229141777392950569846, 6.32674466492762317457414082714, 6.69035056241851935137033607729, 8.076850814550826086680602081307, 10.08954762753121328743171207158, 11.58977057104827798896822653947, 11.98420329156152629396447435166, 13.56889990585964471227607432731, 14.89083059723737574524286398105

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