Properties

Degree 2
Conductor 43
Sign $-0.930 + 0.366i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 49.1i·2-s + (−303. + 175. i)3-s − 1.39e3·4-s + (−4.91e3 + 2.83e3i)5-s + (8.61e3 + 1.49e4i)6-s + (1.64e4 + 9.50e3i)7-s + 1.80e4i·8-s + (3.19e4 − 5.53e4i)9-s + (1.39e5 + 2.41e5i)10-s + 8.11e4·11-s + (4.22e5 − 2.43e5i)12-s + (−3.04e5 + 5.27e5i)13-s + (4.66e5 − 8.08e5i)14-s + (9.95e5 − 1.72e6i)15-s − 5.38e5·16-s + (−1.25e6 + 2.17e6i)17-s + ⋯
L(s)  = 1  − 1.53i·2-s + (−1.24 + 0.721i)3-s − 1.35·4-s + (−1.57 + 0.908i)5-s + (1.10 + 1.91i)6-s + (0.979 + 0.565i)7-s + 0.550i·8-s + (0.540 − 0.936i)9-s + (1.39 + 2.41i)10-s + 0.503·11-s + (1.69 − 0.979i)12-s + (−0.820 + 1.42i)13-s + (0.868 − 1.50i)14-s + (1.31 − 2.27i)15-s − 0.513·16-s + (−0.885 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.930 + 0.366i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.930 + 0.366i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.00130709 - 0.00689390i\)
\(L(\frac12)\)  \(\approx\)  \(0.00130709 - 0.00689390i\)
\(L(6)\)   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 \( 1 + (4.18e7 + 1.40e8i)T \)
good2 \( 1 + 49.1iT - 1.02e3T^{2} \)
3 \( 1 + (303. - 175. i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (4.91e3 - 2.83e3i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (-1.64e4 - 9.50e3i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 - 8.11e4T + 2.59e10T^{2} \)
13 \( 1 + (3.04e5 - 5.27e5i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (1.25e6 - 2.17e6i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (1.81e6 - 1.04e6i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (8.15e5 + 1.41e6i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (-1.79e7 - 1.03e7i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (1.62e7 + 2.81e7i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (7.05e7 - 4.07e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 - 5.61e7T + 1.34e16T^{2} \)
47 \( 1 + 1.07e8T + 5.25e16T^{2} \)
53 \( 1 + (-1.36e8 - 2.35e8i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 - 8.28e8T + 5.11e17T^{2} \)
61 \( 1 + (-2.95e8 - 1.70e8i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (3.39e8 + 5.88e8i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (1.82e9 + 1.05e9i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (1.47e8 + 8.54e7i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (8.89e7 - 1.54e8i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (-1.76e9 - 3.05e9i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (-1.94e9 + 1.12e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 + 1.37e10T + 7.37e19T^{2} \)
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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

−12.24977136758978109289683372367, −11.73255972656429589436919296324, −11.08697417291033387188210133187, −10.35610627805690516195317477632, −8.612480983143022465416701908515, −6.67005077644818323293913856202, −4.52310764943934648273592106992, −3.94421782180349399711777671393, −2.00790407558190368982931461875, −0.00434549508269882356117319248, 0.73314097815727163843937408603, 4.59751676575085770586854155651, 5.20666977834093033319549488638, 6.94715421703849893811597598495, 7.59996547764337848916934887047, 8.606283119018236705575559233780, 11.13060209503661514468544913596, 11.92640052045835886618854687554, 13.05954922518393018804225177674, 14.61513663418073267311291917693

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