Properties

Label 2-43-43.37-c10-0-0
Degree $2$
Conductor $43$
Sign $-0.930 - 0.366i$
Analytic cond. $27.3203$
Root an. cond. $5.22688$
Motivic weight $10$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.930 - 0.366i$
Analytic conductor: \(27.3203\)
Root analytic conductor: \(5.22688\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5),\ -0.930 - 0.366i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.00130709 + 0.00689390i\)
\(L(\frac12)\) \(\approx\) \(0.00130709 + 0.00689390i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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