Properties

Degree 2
Conductor 43
Sign $-0.171 - 0.985i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.2i·2-s + 77.9i·3-s + 129.·4-s + 278. i·5-s + 875.·6-s + 3.16e3i·7-s − 4.33e3i·8-s + 479.·9-s + 3.12e3·10-s − 1.63e4·11-s + 1.01e4i·12-s − 529.·13-s + 3.54e4·14-s − 2.16e4·15-s − 1.53e4·16-s − 1.22e5·17-s + ⋯
L(s)  = 1  − 0.701i·2-s + 0.962i·3-s + 0.507·4-s + 0.444i·5-s + 0.675·6-s + 1.31i·7-s − 1.05i·8-s + 0.0730·9-s + 0.312·10-s − 1.11·11-s + 0.488i·12-s − 0.0185·13-s + 0.923·14-s − 0.428·15-s − 0.234·16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.171 - 0.985i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.171 - 0.985i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.08199 + 1.28688i\)
\(L(\frac12)\)  \(\approx\)  \(1.08199 + 1.28688i\)
\(L(5)\)   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 \( 1 + (5.86e5 + 3.36e6i)T \)
good2 \( 1 + 11.2iT - 256T^{2} \)
3 \( 1 - 77.9iT - 6.56e3T^{2} \)
5 \( 1 - 278. iT - 3.90e5T^{2} \)
7 \( 1 - 3.16e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.63e4T + 2.14e8T^{2} \)
13 \( 1 + 529.T + 8.15e8T^{2} \)
17 \( 1 + 1.22e5T + 6.97e9T^{2} \)
19 \( 1 - 1.26e5iT - 1.69e10T^{2} \)
23 \( 1 + 3.32e4T + 7.83e10T^{2} \)
29 \( 1 - 1.14e6iT - 5.00e11T^{2} \)
31 \( 1 - 7.37e5T + 8.52e11T^{2} \)
37 \( 1 - 1.37e6iT - 3.51e12T^{2} \)
41 \( 1 + 5.06e6T + 7.98e12T^{2} \)
47 \( 1 + 3.40e6T + 2.38e13T^{2} \)
53 \( 1 - 8.04e6T + 6.22e13T^{2} \)
59 \( 1 - 1.77e7T + 1.46e14T^{2} \)
61 \( 1 + 8.79e6iT - 1.91e14T^{2} \)
67 \( 1 - 2.55e6T + 4.06e14T^{2} \)
71 \( 1 - 1.00e7iT - 6.45e14T^{2} \)
73 \( 1 + 7.54e6iT - 8.06e14T^{2} \)
79 \( 1 - 1.30e7T + 1.51e15T^{2} \)
83 \( 1 - 7.73e7T + 2.25e15T^{2} \)
89 \( 1 - 3.53e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.12e8T + 7.83e15T^{2} \)
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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

−14.98851410617753441456478708159, −13.05345659558603419440903888000, −11.95469865499922234084605751914, −10.77297328146748040139490777818, −10.07020674262189388443675470914, −8.663589772848962489218868004827, −6.74030582042873742021249193014, −5.09588201444298230055405919432, −3.30964323049856846357641116668, −2.11551654132909601522400149813, 0.59520557299264531433888982469, 2.26635127261639861780204296217, 4.71340525251480943534356723244, 6.52722608175218914698479468641, 7.31364937395933833907658110037, 8.312729079154963208812443697602, 10.36422749828244714568014600737, 11.54595688486700018601911812657, 13.12833386931052635991839429037, 13.58839754008003014752885881654

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