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Properties

Label 2-43-43.7-c4-0-4
Degree $2$
Conductor $43$
Sign $-0.488 - 0.872i$
Analytic cond. $4.44490$
Root an. cond. $2.10829$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.83i·2-s + (7.61 − 4.39i)3-s − 7.40·4-s + (−37.8 + 21.8i)5-s + (21.2 + 36.8i)6-s + (66.6 + 38.4i)7-s + 41.5i·8-s + (−1.82 + 3.15i)9-s + (−105. − 183. i)10-s − 12.1·11-s + (−56.4 + 32.5i)12-s + (58.5 − 101. i)13-s + (−186. + 322. i)14-s + (−192. + 332. i)15-s − 319.·16-s + (171. − 296. i)17-s + ⋯
L(s)  = 1  + 1.20i·2-s + (0.846 − 0.488i)3-s − 0.463·4-s + (−1.51 + 0.873i)5-s + (0.591 + 1.02i)6-s + (1.36 + 0.785i)7-s + 0.649i·8-s + (−0.0224 + 0.0389i)9-s + (−1.05 − 1.83i)10-s − 0.100·11-s + (−0.391 + 0.226i)12-s + (0.346 − 0.600i)13-s + (−0.949 + 1.64i)14-s + (−0.853 + 1.47i)15-s − 1.24·16-s + (0.592 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(4.44490\)
Root analytic conductor: \(2.10829\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :2),\ -0.488 - 0.872i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.880195 + 1.50117i\)
\(L(\frac12)\) \(\approx\) \(0.880195 + 1.50117i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.68e3 - 760. i)T \)
good2 \( 1 - 4.83iT - 16T^{2} \)
3 \( 1 + (-7.61 + 4.39i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (37.8 - 21.8i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-66.6 - 38.4i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + 12.1T + 1.46e4T^{2} \)
13 \( 1 + (-58.5 + 101. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-171. + 296. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-280. + 161. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (168. + 291. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-369. - 213. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-331. - 573. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-1.76e3 + 1.01e3i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 2.82e3T + 2.82e6T^{2} \)
47 \( 1 - 2.64e3T + 4.87e6T^{2} \)
53 \( 1 + (1.37e3 + 2.38e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 - 3.92e3T + 1.21e7T^{2} \)
61 \( 1 + (-664. - 383. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (2.74e3 + 4.74e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-995. - 574. i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (1.62e3 + 939. i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (2.60e3 - 4.51e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-2.30e3 - 3.98e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (7.10e3 - 4.10e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 1.61e3T + 8.85e7T^{2} \)
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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

−15.34701302623563644705515883219, −14.68611168470395883972653374935, −13.95339017461122590190334088467, −11.90510989928767373042057096137, −11.09840994587811215068801906977, −8.456964519981592222758928419942, −7.953649634307607280241190253176, −7.09190167330039891808053106137, −5.11296679822685808870245711818, −2.79877851973951934840416443240, 1.21594991144697204885437831844, 3.63092513689372295152178809249, 4.37766373545764783113278227663, 7.73845612753024017241686189025, 8.601234584262162857266728005095, 10.11051114803154418441718666867, 11.48806329535333397761302615344, 11.96689076656101029399582178802, 13.52616463562220135232714421449, 14.81259337828775013426764646346

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