Properties

Label 2-43-43.18-c6-0-2
Degree $2$
Conductor $43$
Sign $-0.915 - 0.403i$
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  + (0.965 − 2.00i)2-s + (−22.0 + 32.3i)3-s + (36.8 + 46.1i)4-s + (89.0 + 96.0i)5-s + (43.6 + 75.5i)6-s + (−260. − 150. i)7-s + (267. − 60.9i)8-s + (−294. − 751. i)9-s + (278. − 85.9i)10-s + (−688. + 863. i)11-s + (−2.30e3 + 172. i)12-s + (−194. − 59.9i)13-s + (−552. + 376. i)14-s + (−5.07e3 + 765. i)15-s + (−705. + 3.08e3i)16-s + (−2.89e3 − 2.68e3i)17-s + ⋯
L(s)  = 1  + (0.120 − 0.250i)2-s + (−0.817 + 1.19i)3-s + (0.575 + 0.721i)4-s + (0.712 + 0.768i)5-s + (0.201 + 0.349i)6-s + (−0.758 − 0.437i)7-s + (0.521 − 0.119i)8-s + (−0.404 − 1.03i)9-s + (0.278 − 0.0859i)10-s + (−0.517 + 0.648i)11-s + (−1.33 + 0.100i)12-s + (−0.0885 − 0.0272i)13-s + (−0.201 + 0.137i)14-s + (−1.50 + 0.226i)15-s + (−0.172 + 0.754i)16-s + (−0.589 − 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.915 - 0.403i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ -0.915 - 0.403i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.263214 + 1.25088i\)
\(L(\frac12)\) \(\approx\) \(0.263214 + 1.25088i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-6.67e4 - 4.32e4i)T \)
good2 \( 1 + (-0.965 + 2.00i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (22.0 - 32.3i)T + (-266. - 678. i)T^{2} \)
5 \( 1 + (-89.0 - 96.0i)T + (-1.16e3 + 1.55e4i)T^{2} \)
7 \( 1 + (260. + 150. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (688. - 863. i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (194. + 59.9i)T + (3.98e6 + 2.71e6i)T^{2} \)
17 \( 1 + (2.89e3 + 2.68e3i)T + (1.80e6 + 2.40e7i)T^{2} \)
19 \( 1 + (-6.01e3 - 2.36e3i)T + (3.44e7 + 3.19e7i)T^{2} \)
23 \( 1 + (9.94e3 + 1.49e3i)T + (1.41e8 + 4.36e7i)T^{2} \)
29 \( 1 + (8.25e3 + 1.21e4i)T + (-2.17e8 + 5.53e8i)T^{2} \)
31 \( 1 + (-1.51e3 - 2.01e4i)T + (-8.77e8 + 1.32e8i)T^{2} \)
37 \( 1 + (2.20e4 - 1.27e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + (-1.10e5 - 5.32e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-1.26e5 - 1.58e5i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-8.39e4 + 2.58e4i)T + (1.83e10 - 1.24e10i)T^{2} \)
59 \( 1 + (3.84e4 - 1.68e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (1.64e5 + 1.23e4i)T + (5.09e10 + 7.67e9i)T^{2} \)
67 \( 1 + (-9.62e4 + 2.45e5i)T + (-6.63e10 - 6.15e10i)T^{2} \)
71 \( 1 + (-5.97e4 - 3.96e5i)T + (-1.22e11 + 3.77e10i)T^{2} \)
73 \( 1 + (1.53e5 - 4.99e5i)T + (-1.25e11 - 8.52e10i)T^{2} \)
79 \( 1 + (3.13e5 - 5.43e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (4.21e5 + 2.87e5i)T + (1.19e11 + 3.04e11i)T^{2} \)
89 \( 1 + (340. - 499. i)T + (-1.81e11 - 4.62e11i)T^{2} \)
97 \( 1 + (-8.80e5 + 1.10e6i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−15.64206735521130408180395968143, −14.02463135635929319486115283807, −12.67016857867335143691725231578, −11.36616021484959668172358051092, −10.40637055553300889305200240464, −9.689643087610467794553138712648, −7.32949320375483878851524024289, −6.01674083508771214111288402920, −4.26019514176819167642144793150, −2.73338693879746506746256180825, 0.61022115054036160741542929339, 2.05389140312083764157768687676, 5.52778954122683748938399047612, 6.04948502850950988876805484802, 7.36006222926031432536887779378, 9.233218628807355799976730426254, 10.73076867976400634108411829116, 11.99905072062793464117574499719, 13.02402911663668373404745288806, 13.83712107732843534345309648998

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