Properties

Label 2-43-43.36-c5-0-16
Degree $2$
Conductor $43$
Sign $-0.0207 - 0.999i$
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  − 3.30·2-s + (−12.3 − 21.4i)3-s − 21.0·4-s + (−32.1 − 55.6i)5-s + (40.8 + 70.8i)6-s + (−2.83 + 4.90i)7-s + 175.·8-s + (−183. + 318. i)9-s + (106. + 184. i)10-s − 20.0·11-s + (260. + 450. i)12-s + (308. − 534. i)13-s + (9.37 − 16.2i)14-s + (−794. + 1.37e3i)15-s + 92.8·16-s + (−987. + 1.71e3i)17-s + ⋯
L(s)  = 1  − 0.584·2-s + (−0.792 − 1.37i)3-s − 0.657·4-s + (−0.574 − 0.995i)5-s + (0.463 + 0.803i)6-s + (−0.0218 + 0.0378i)7-s + 0.969·8-s + (−0.757 + 1.31i)9-s + (0.336 + 0.582i)10-s − 0.0498·11-s + (0.521 + 0.903i)12-s + (0.505 − 0.876i)13-s + (0.0127 − 0.0221i)14-s + (−0.911 + 1.57i)15-s + 0.0906·16-s + (−0.829 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0207 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0207 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.0207 - 0.999i$
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: \((2,\ 43,\ (\ :5/2),\ -0.0207 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0735491 + 0.0750937i\)
\(L(\frac12)\) \(\approx\) \(0.0735491 + 0.0750937i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (8.00e3 - 9.10e3i)T \)
good2 \( 1 + 3.30T + 32T^{2} \)
3 \( 1 + (12.3 + 21.4i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (32.1 + 55.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (2.83 - 4.90i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 20.0T + 1.61e5T^{2} \)
13 \( 1 + (-308. + 534. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (987. - 1.71e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (488. + 846. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (165. + 286. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-4.08e3 + 7.08e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.16e3 - 3.74e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-3.70e3 - 6.42e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.89e4T + 1.15e8T^{2} \)
47 \( 1 + 7.98e3T + 2.29e8T^{2} \)
53 \( 1 + (-1.12e4 - 1.95e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + 2.25e4T + 7.14e8T^{2} \)
61 \( 1 + (-1.05e3 + 1.82e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.11e4 + 1.93e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-2.48e3 + 4.30e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (1.63e4 - 2.83e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.63e4 + 4.55e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (5.83e4 + 1.01e5i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-1.93e4 - 3.35e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 7.58e4T + 8.58e9T^{2} \)
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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

−13.45217733045053363112458551893, −12.92761005602917417101053622686, −11.89912734500939927924946286845, −10.49201222566104584962033428328, −8.579136707856064003433488037778, −7.965358596766216702184794005796, −6.26281057406068035469109715874, −4.63833533553024383296019160391, −1.24135982214275932262921091854, −0.090676706930431585445166943626, 3.75372048749109356513661220368, 4.96195122090520505746854662481, 6.89754126995870905153465005004, 8.751136621549814788828452014961, 9.898078225416571480894647937990, 10.81796490663938994219724375303, 11.67783464226615999801243762388, 13.74702965492259254390468392181, 14.91665849235111724010410244615, 16.02676910934446679744100229159

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