Properties

Label 2-43-43.6-c5-0-5
Degree $2$
Conductor $43$
Sign $0.621 - 0.783i$
Analytic cond. $6.89650$
Root an. cond. $2.62611$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.45·2-s + (9.81 − 16.9i)3-s − 12.1·4-s + (−39.8 + 69.0i)5-s + (−43.6 + 75.6i)6-s + (8.78 + 15.2i)7-s + 196.·8-s + (−71.0 − 123. i)9-s + (177. − 307. i)10-s + 296.·11-s + (−119. + 207. i)12-s + (198. + 343. i)13-s + (−39.1 − 67.7i)14-s + (782. + 1.35e3i)15-s − 485.·16-s + (676. + 1.17e3i)17-s + ⋯
L(s)  = 1  − 0.786·2-s + (0.629 − 1.09i)3-s − 0.380·4-s + (−0.713 + 1.23i)5-s + (−0.495 + 0.857i)6-s + (0.0677 + 0.117i)7-s + 1.08·8-s + (−0.292 − 0.506i)9-s + (0.561 − 0.972i)10-s + 0.738·11-s + (−0.239 + 0.415i)12-s + (0.325 + 0.563i)13-s + (−0.0533 − 0.0923i)14-s + (0.898 + 1.55i)15-s − 0.474·16-s + (0.567 + 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.831073 + 0.401426i\)
\(L(\frac12)\) \(\approx\) \(0.831073 + 0.401426i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-865. - 1.20e4i)T \)
good2 \( 1 + 4.45T + 32T^{2} \)
3 \( 1 + (-9.81 + 16.9i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (39.8 - 69.0i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-8.78 - 15.2i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 296.T + 1.61e5T^{2} \)
13 \( 1 + (-198. - 343. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-676. - 1.17e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (680. - 1.17e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (570. - 987. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.19e3 + 2.07e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (2.83e3 - 4.91e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (3.03e3 - 5.26e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.04e4T + 1.15e8T^{2} \)
47 \( 1 + 2.68e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.51e4 + 2.62e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + 3.00e3T + 7.14e8T^{2} \)
61 \( 1 + (6.41e3 + 1.11e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.94e4 - 3.37e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-2.31e4 - 4.01e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-1.48e4 - 2.57e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.40e4 - 5.89e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.83e4 + 6.64e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-6.45e4 + 1.11e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 5.70e4T + 8.58e9T^{2} \)
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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

−14.74232612988920559545054143902, −14.12884878654671909548817299140, −12.84414654224267357240513158164, −11.44678855004943839230644529932, −10.13614138587058482110914705761, −8.564075538766958881318868029931, −7.70876862262568875575860493283, −6.62373306572772777144431686258, −3.71201309120114240770580918673, −1.62549367191514006370273365810, 0.67875054881750469881790122093, 3.88437045429982784314069203981, 4.88248304534816697803091476242, 7.76997190502498829849944331772, 8.915373714387469585941932100035, 9.375901133675999276786151866067, 10.77859529314794697274423243539, 12.38415923541474853569794091063, 13.74168541795648646105243028118, 15.02304194038131780092239215529

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