Properties

Label 2-43-43.36-c5-0-0
Degree $2$
Conductor $43$
Sign $-0.880 - 0.473i$
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  + 1.74·2-s + (7.37 + 12.7i)3-s − 28.9·4-s + (−8.29 − 14.3i)5-s + (12.8 + 22.2i)6-s + (−98.7 + 171. i)7-s − 106.·8-s + (12.6 − 21.9i)9-s + (−14.4 − 25.0i)10-s − 376.·11-s + (−213. − 369. i)12-s + (−55.5 + 96.2i)13-s + (−172. + 298. i)14-s + (122. − 212. i)15-s + 741.·16-s + (−786. + 1.36e3i)17-s + ⋯
L(s)  = 1  + 0.308·2-s + (0.473 + 0.819i)3-s − 0.904·4-s + (−0.148 − 0.257i)5-s + (0.145 + 0.252i)6-s + (−0.761 + 1.31i)7-s − 0.587·8-s + (0.0522 − 0.0905i)9-s + (−0.0457 − 0.0792i)10-s − 0.938·11-s + (−0.428 − 0.741i)12-s + (−0.0911 + 0.157i)13-s + (−0.234 + 0.406i)14-s + (0.140 − 0.243i)15-s + 0.724·16-s + (−0.659 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ -0.880 - 0.473i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.232480 + 0.923063i\)
\(L(\frac12)\) \(\approx\) \(0.232480 + 0.923063i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-3.98e3 - 1.14e4i)T \)
good2 \( 1 - 1.74T + 32T^{2} \)
3 \( 1 + (-7.37 - 12.7i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (8.29 + 14.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (98.7 - 171. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 376.T + 1.61e5T^{2} \)
13 \( 1 + (55.5 - 96.2i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (786. - 1.36e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-444. - 769. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-428. - 741. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-12.2 + 21.1i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.22e3 - 3.84e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (3.15e3 + 5.46e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 5.20e3T + 1.15e8T^{2} \)
47 \( 1 + 1.33e4T + 2.29e8T^{2} \)
53 \( 1 + (-2.12e3 - 3.68e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + 4.94e4T + 7.14e8T^{2} \)
61 \( 1 + (1.70e4 - 2.94e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.10e4 - 3.64e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (1.94e4 - 3.36e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (1.27e4 - 2.20e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.78e4 + 8.29e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.03e3 + 1.79e3i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (7.30e4 + 1.26e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.89e4T + 8.58e9T^{2} \)
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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

−15.39000819382786241839860120024, −14.46423415403037350463915050062, −12.99030867393103136004296009879, −12.30086730919340604289909357285, −10.26580948379974427649203343163, −9.206235404802995205829774150417, −8.437542810401349719793447789495, −5.93936188653486540196539228417, −4.53665374848196007532433372082, −3.08157456964692314733226017521, 0.44777139808328150145092053179, 3.06404873643364958740461839416, 4.77747795841552057659401376132, 6.88875152960968160232838396741, 7.888947050339586659933869448802, 9.471413809022138950838475263371, 10.74690916954533346047120154607, 12.64017619720716390962755685047, 13.50858409967213007631408748345, 13.85327147232206958349660824521

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