Properties

Label 2-43-43.37-c6-0-19
Degree $2$
Conductor $43$
Sign $0.0341 - 0.999i$
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  − 15.5i·2-s + (25.2 + 14.5i)3-s − 177.·4-s + (−105. − 61.0i)5-s + (226. − 393. i)6-s + (−10.5 + 6.07i)7-s + 1.77e3i·8-s + (61.0 + 105. i)9-s + (−949. + 1.64e3i)10-s − 1.76e3·11-s + (−4.49e3 − 2.59e3i)12-s + (1.47e3 + 2.55e3i)13-s + (94.4 + 163. i)14-s + (−1.78e3 − 3.08e3i)15-s + 1.61e4·16-s + (397. + 689. i)17-s + ⋯
L(s)  = 1  − 1.94i·2-s + (0.935 + 0.540i)3-s − 2.78·4-s + (−0.845 − 0.488i)5-s + (1.05 − 1.81i)6-s + (−0.0306 + 0.0177i)7-s + 3.46i·8-s + (0.0837 + 0.144i)9-s + (−0.949 + 1.64i)10-s − 1.32·11-s + (−2.60 − 1.50i)12-s + (0.671 + 1.16i)13-s + (0.0344 + 0.0596i)14-s + (−0.527 − 0.913i)15-s + 3.95·16-s + (0.0809 + 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0341 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0341 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.0341 - 0.999i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ 0.0341 - 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.253949 + 0.245414i\)
\(L(\frac12)\) \(\approx\) \(0.253949 + 0.245414i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (7.93e4 - 4.13e3i)T \)
good2 \( 1 + 15.5iT - 64T^{2} \)
3 \( 1 + (-25.2 - 14.5i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (105. + 61.0i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (10.5 - 6.07i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + 1.76e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.47e3 - 2.55e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (-397. - 689. i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (1.06e4 + 6.13e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-2.84e3 + 4.92e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-531. + 306. i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (3.71e3 - 6.43e3i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (8.67e4 + 5.00e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 2.85e4T + 4.75e9T^{2} \)
47 \( 1 - 6.53e3T + 1.07e10T^{2} \)
53 \( 1 + (-1.95e3 + 3.37e3i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 - 3.51e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.05e5 + 1.18e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-2.02e4 + 3.51e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (2.64e5 - 1.52e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (1.37e5 - 7.92e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.05e5 + 1.82e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-4.75e5 + 8.22e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-5.58e5 - 3.22e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 1.20e6T + 8.32e11T^{2} \)
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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

−13.44821459609410795428234941797, −12.51163548196310644091713417347, −11.31353833338047674710844150326, −10.30645658270386146186758870454, −8.922549033401082592923060423205, −8.404878508861011715803402391460, −4.64430664016920665935045490757, −3.68400902349200347745133217541, −2.29090438842916028662009437939, −0.14678293376653745041513948080, 3.51376556126503208884351626061, 5.41823854488809662225413413938, 7.00071080590927172928035952729, 8.062960892167784214244638473189, 8.397597980588770584857484022097, 10.36020998837000991920648449021, 12.89111729684651581603764092555, 13.54185432929600222167459575230, 14.83675584283037260692884728069, 15.31065225150218807919657361117

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