Properties

Label 2-43-43.12-c6-0-20
Degree $2$
Conductor $43$
Sign $0.980 + 0.196i$
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  + (−6.35 − 13.1i)2-s + (−28.5 − 41.8i)3-s + (−93.6 + 117. i)4-s + (6.53 − 7.04i)5-s + (−370. + 642. i)6-s + (234. − 135. i)7-s + (1.23e3 + 280. i)8-s + (−671. + 1.71e3i)9-s + (−134. − 41.4i)10-s + (−1.49e3 − 1.87e3i)11-s + (7.59e3 + 568. i)12-s + (−2.46e3 + 761. i)13-s + (−3.27e3 − 2.23e3i)14-s + (−481. − 72.5i)15-s + (−1.97e3 − 8.63e3i)16-s + (1.31e3 − 1.22e3i)17-s + ⋯
L(s)  = 1  + (−0.793 − 1.64i)2-s + (−1.05 − 1.55i)3-s + (−1.46 + 1.83i)4-s + (0.0522 − 0.0563i)5-s + (−1.71 + 2.97i)6-s + (0.684 − 0.395i)7-s + (2.40 + 0.548i)8-s + (−0.921 + 2.34i)9-s + (−0.134 − 0.0414i)10-s + (−1.12 − 1.41i)11-s + (4.39 + 0.329i)12-s + (−1.12 + 0.346i)13-s + (−1.19 − 0.814i)14-s + (−0.142 − 0.0214i)15-s + (−0.481 − 2.10i)16-s + (0.267 − 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.980 + 0.196i$
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: \((2,\ 43,\ (\ :3),\ 0.980 + 0.196i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0815430 - 0.00807628i\)
\(L(\frac12)\) \(\approx\) \(0.0815430 - 0.00807628i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (7.94e4 + 2.95e3i)T \)
good2 \( 1 + (6.35 + 13.1i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (28.5 + 41.8i)T + (-266. + 678. i)T^{2} \)
5 \( 1 + (-6.53 + 7.04i)T + (-1.16e3 - 1.55e4i)T^{2} \)
7 \( 1 + (-234. + 135. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (1.49e3 + 1.87e3i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (2.46e3 - 761. i)T + (3.98e6 - 2.71e6i)T^{2} \)
17 \( 1 + (-1.31e3 + 1.22e3i)T + (1.80e6 - 2.40e7i)T^{2} \)
19 \( 1 + (-2.37e3 + 931. i)T + (3.44e7 - 3.19e7i)T^{2} \)
23 \( 1 + (-4.21e3 + 635. i)T + (1.41e8 - 4.36e7i)T^{2} \)
29 \( 1 + (-1.49e4 + 2.18e4i)T + (-2.17e8 - 5.53e8i)T^{2} \)
31 \( 1 + (-1.31e3 + 1.76e4i)T + (-8.77e8 - 1.32e8i)T^{2} \)
37 \( 1 + (3.19e4 + 1.84e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (8.39e4 - 4.04e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-9.43e3 + 1.18e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (-2.64e5 - 8.17e4i)T + (1.83e10 + 1.24e10i)T^{2} \)
59 \( 1 + (-3.79e4 - 1.66e5i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (-1.49e5 + 1.12e4i)T + (5.09e10 - 7.67e9i)T^{2} \)
67 \( 1 + (-5.97e4 - 1.52e5i)T + (-6.63e10 + 6.15e10i)T^{2} \)
71 \( 1 + (7.34e3 - 4.87e4i)T + (-1.22e11 - 3.77e10i)T^{2} \)
73 \( 1 + (-2.48e4 - 8.04e4i)T + (-1.25e11 + 8.52e10i)T^{2} \)
79 \( 1 + (1.25e5 + 2.17e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (5.55e5 - 3.78e5i)T + (1.19e11 - 3.04e11i)T^{2} \)
89 \( 1 + (1.43e5 + 2.09e5i)T + (-1.81e11 + 4.62e11i)T^{2} \)
97 \( 1 + (-1.80e5 - 2.26e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−13.09667041696197543089987502950, −11.86337009211895785874933235757, −11.33898270806453692431370353957, −10.31237629421529955080713438056, −8.361631904578489822548629308007, −7.40312394544256056582966434733, −5.22854544222061443225760595498, −2.55741661670586541983443182585, −1.13216083318856740717370445919, −0.06793021863786863003389950434, 4.94581470557018831615992692559, 5.22180644007470487356957953678, 6.94314253287539266175040483937, 8.477669111668260871277858953878, 9.939277478485765890701013962065, 10.31627670985035319325315570739, 12.15105793894405070545872667121, 14.57050726474985445532471750871, 15.17709379527768878619920410323, 15.87061516002831456999689877685

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