Properties

Label 2-43-43.37-c6-0-4
Degree $2$
Conductor $43$
Sign $-0.175 + 0.984i$
Analytic cond. $9.89232$
Root an. cond. $3.14520$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.1i·2-s + (15.3 + 8.86i)3-s − 165.·4-s + (63.9 + 36.9i)5-s + (−134. + 232. i)6-s + (−298. + 172. i)7-s − 1.53e3i·8-s + (−207. − 358. i)9-s + (−559. + 969. i)10-s − 1.17e3·11-s + (−2.54e3 − 1.46e3i)12-s + (642. + 1.11e3i)13-s + (−2.60e3 − 4.51e3i)14-s + (655. + 1.13e3i)15-s + 1.27e4·16-s + (1.58e3 + 2.74e3i)17-s + ⋯
L(s)  = 1  + 1.89i·2-s + (0.568 + 0.328i)3-s − 2.58·4-s + (0.511 + 0.295i)5-s + (−0.622 + 1.07i)6-s + (−0.869 + 0.501i)7-s − 3.00i·8-s + (−0.284 − 0.492i)9-s + (−0.559 + 0.969i)10-s − 0.880·11-s + (−1.47 − 0.849i)12-s + (0.292 + 0.506i)13-s + (−0.950 − 1.64i)14-s + (0.194 + 0.336i)15-s + 3.10·16-s + (0.323 + 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.175 + 0.984i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ -0.175 + 0.984i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.658881 - 0.786493i\)
\(L(\frac12)\) \(\approx\) \(0.658881 - 0.786493i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-7.91e4 - 7.13e3i)T \)
good2 \( 1 - 15.1iT - 64T^{2} \)
3 \( 1 + (-15.3 - 8.86i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (-63.9 - 36.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (298. - 172. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + 1.17e3T + 1.77e6T^{2} \)
13 \( 1 + (-642. - 1.11e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (-1.58e3 - 2.74e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-5.76e3 - 3.33e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (7.35e3 - 1.27e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-3.26e3 + 1.88e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (2.91e4 - 5.05e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (2.24e4 + 1.29e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 2.69e4T + 4.75e9T^{2} \)
47 \( 1 + 1.98e5T + 1.07e10T^{2} \)
53 \( 1 + (1.20e5 - 2.08e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 - 1.63e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.66e5 + 1.53e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.45e5 - 2.52e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-5.00e5 + 2.89e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (-9.58e3 + 5.53e3i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-3.48e5 - 6.04e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-9.46e4 + 1.63e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-4.23e5 - 2.44e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 8.95e5T + 8.32e11T^{2} \)
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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

−15.73715755245099110918048147470, −14.49563821687218419054382979669, −13.85778259776621702502833432869, −12.59353491202717870446752315660, −9.923838844983031787020296564451, −9.129102817679951965554664993131, −7.933260601821586349137050051053, −6.45313656296827348355994421742, −5.52285971338927144737666415465, −3.52063714639869336309684566954, 0.44637450956478186238126633273, 2.21892293220381868457873886964, 3.36517977798371795820027025574, 5.24042079799651680655904116058, 7.966251035491587050059962415139, 9.354674828932611328845388424820, 10.21427057927423053581266870416, 11.36659944102517265576533321291, 13.01185503998995896269093596164, 13.19498770571592039644241633947

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