Properties

Degree 2
Conductor 43
Sign $-0.0437 - 0.999i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.4i·2-s + 49.3i·3-s + 47.3·4-s − 738. i·5-s − 712.·6-s + 198. i·7-s + 4.38e3i·8-s + 4.12e3·9-s + 1.06e4·10-s + 1.24e4·11-s + 2.33e3i·12-s + 2.78e4·13-s − 2.86e3·14-s + 3.64e4·15-s − 5.11e4·16-s − 6.64e4·17-s + ⋯
L(s)  = 1  + 0.902i·2-s + 0.609i·3-s + 0.184·4-s − 1.18i·5-s − 0.549·6-s + 0.0826i·7-s + 1.06i·8-s + 0.629·9-s + 1.06·10-s + 0.849·11-s + 0.112i·12-s + 0.974·13-s − 0.0746·14-s + 0.719·15-s − 0.781·16-s − 0.795·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0437 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0437 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.0437 - 0.999i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.0437 - 0.999i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.65695 + 1.73106i\)
\(L(\frac12)\)  \(\approx\)  \(1.65695 + 1.73106i\)
\(L(5)\)   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 \( 1 + (1.49e5 + 3.41e6i)T \)
good2 \( 1 - 14.4iT - 256T^{2} \)
3 \( 1 - 49.3iT - 6.56e3T^{2} \)
5 \( 1 + 738. iT - 3.90e5T^{2} \)
7 \( 1 - 198. iT - 5.76e6T^{2} \)
11 \( 1 - 1.24e4T + 2.14e8T^{2} \)
13 \( 1 - 2.78e4T + 8.15e8T^{2} \)
17 \( 1 + 6.64e4T + 6.97e9T^{2} \)
19 \( 1 - 1.54e5iT - 1.69e10T^{2} \)
23 \( 1 - 2.53e5T + 7.83e10T^{2} \)
29 \( 1 + 3.32e5iT - 5.00e11T^{2} \)
31 \( 1 + 7.16e5T + 8.52e11T^{2} \)
37 \( 1 - 2.30e6iT - 3.51e12T^{2} \)
41 \( 1 + 1.26e5T + 7.98e12T^{2} \)
47 \( 1 - 3.57e6T + 2.38e13T^{2} \)
53 \( 1 + 4.88e6T + 6.22e13T^{2} \)
59 \( 1 - 5.89e6T + 1.46e14T^{2} \)
61 \( 1 + 3.89e6iT - 1.91e14T^{2} \)
67 \( 1 - 2.99e7T + 4.06e14T^{2} \)
71 \( 1 - 4.99e6iT - 6.45e14T^{2} \)
73 \( 1 + 1.46e7iT - 8.06e14T^{2} \)
79 \( 1 + 1.87e7T + 1.51e15T^{2} \)
83 \( 1 - 7.06e7T + 2.25e15T^{2} \)
89 \( 1 + 1.63e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.07e8T + 7.83e15T^{2} \)
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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

−14.91216758210998081367655266451, −13.50564395533573427527470657893, −12.20432180730008424662693537295, −10.87444899833908303772257813982, −9.262294687585012302136681825716, −8.326750103906845158650565845145, −6.74617695005928348121407227318, −5.37875039537024956748854667721, −4.06258067852579040687085557775, −1.44396101608817631505358528992, 1.08888187820332121352924683995, 2.46002686138896517816116517553, 3.84555748014152724487315649467, 6.56711375415879341471364632252, 7.12371833633735312736146124275, 9.237334428137844785872540402609, 10.75923375755566151870536236950, 11.26240979429042131263574358657, 12.63572078557354469218888593495, 13.59724178135724369422502704966

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