Properties

Label 2-43-43.12-c6-0-14
Degree $2$
Conductor $43$
Sign $0.605 + 0.796i$
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  + (−0.799 − 1.66i)2-s + (12.6 + 18.5i)3-s + (37.7 − 47.3i)4-s + (102. − 110. i)5-s + (20.7 − 35.9i)6-s + (−237. + 137. i)7-s + (−223. − 51.1i)8-s + (81.3 − 207. i)9-s + (−265. − 81.9i)10-s + (−158. − 198. i)11-s + (1.35e3 + 101. i)12-s + (2.08e3 − 642. i)13-s + (418. + 285. i)14-s + (3.35e3 + 505. i)15-s + (−768. − 3.36e3i)16-s + (2.49e3 − 2.31e3i)17-s + ⋯
L(s)  = 1  + (−0.0999 − 0.207i)2-s + (0.469 + 0.688i)3-s + (0.590 − 0.740i)4-s + (0.820 − 0.884i)5-s + (0.0960 − 0.166i)6-s + (−0.693 + 0.400i)7-s + (−0.437 − 0.0998i)8-s + (0.111 − 0.284i)9-s + (−0.265 − 0.0819i)10-s + (−0.119 − 0.149i)11-s + (0.786 + 0.0589i)12-s + (0.948 − 0.292i)13-s + (0.152 + 0.103i)14-s + (0.994 + 0.149i)15-s + (−0.187 − 0.822i)16-s + (0.508 − 0.471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.605 + 0.796i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ 0.605 + 0.796i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.06523 - 1.02444i\)
\(L(\frac12)\) \(\approx\) \(2.06523 - 1.02444i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (5.75e4 + 5.48e4i)T \)
good2 \( 1 + (0.799 + 1.66i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (-12.6 - 18.5i)T + (-266. + 678. i)T^{2} \)
5 \( 1 + (-102. + 110. i)T + (-1.16e3 - 1.55e4i)T^{2} \)
7 \( 1 + (237. - 137. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (158. + 198. i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (-2.08e3 + 642. i)T + (3.98e6 - 2.71e6i)T^{2} \)
17 \( 1 + (-2.49e3 + 2.31e3i)T + (1.80e6 - 2.40e7i)T^{2} \)
19 \( 1 + (937. - 368. i)T + (3.44e7 - 3.19e7i)T^{2} \)
23 \( 1 + (-6.91e3 + 1.04e3i)T + (1.41e8 - 4.36e7i)T^{2} \)
29 \( 1 + (-2.49e3 + 3.66e3i)T + (-2.17e8 - 5.53e8i)T^{2} \)
31 \( 1 + (3.52e3 - 4.70e4i)T + (-8.77e8 - 1.32e8i)T^{2} \)
37 \( 1 + (-2.70e3 - 1.56e3i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (3.08e4 - 1.48e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (8.16e4 - 1.02e5i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (1.72e4 + 5.31e3i)T + (1.83e10 + 1.24e10i)T^{2} \)
59 \( 1 + (-8.04e4 - 3.52e5i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (-1.16e5 + 8.73e3i)T + (5.09e10 - 7.67e9i)T^{2} \)
67 \( 1 + (2.45e4 + 6.25e4i)T + (-6.63e10 + 6.15e10i)T^{2} \)
71 \( 1 + (1.15e4 - 7.67e4i)T + (-1.22e11 - 3.77e10i)T^{2} \)
73 \( 1 + (1.17e5 + 3.79e5i)T + (-1.25e11 + 8.52e10i)T^{2} \)
79 \( 1 + (49.1 + 85.1i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-4.10e4 + 2.79e4i)T + (1.19e11 - 3.04e11i)T^{2} \)
89 \( 1 + (-7.20e5 - 1.05e6i)T + (-1.81e11 + 4.62e11i)T^{2} \)
97 \( 1 + (-2.58e5 - 3.24e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−14.67043748594800262417421859409, −13.38977345053916435775422271474, −12.17682675366157470454423844912, −10.55410371592537778561970723896, −9.585084556664655861120147688739, −8.819340870288468165185616056504, −6.44027836075738454654481858820, −5.23753151044563546955985079264, −3.10232024073229452844686942649, −1.18710425004753470685180977138, 2.01516846731119690710634820318, 3.34371153290642944030924791333, 6.27539306688324960444307911631, 7.09870349556083099755766064187, 8.324306040884656228501006683174, 10.02297799385498118119615956582, 11.24355675990283261757934238612, 12.88656270747820011366552153853, 13.50633017564577620470211148385, 14.75945494826695020025120823760

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