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Properties

Label 2-43-43.36-c5-0-11
Degree $2$
Conductor $43$
Sign $-0.653 - 0.756i$
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  − 8.35·2-s + (−3.42 − 5.92i)3-s + 37.8·4-s + (−28.3 − 49.1i)5-s + (28.6 + 49.5i)6-s + (46.8 − 81.1i)7-s − 48.8·8-s + (98.0 − 169. i)9-s + (237. + 410. i)10-s − 458.·11-s + (−129. − 224. i)12-s + (−575. + 995. i)13-s + (−391. + 678. i)14-s + (−194. + 336. i)15-s − 802.·16-s + (130. − 225. i)17-s + ⋯
L(s)  = 1  − 1.47·2-s + (−0.219 − 0.380i)3-s + 1.18·4-s + (−0.507 − 0.879i)5-s + (0.324 + 0.561i)6-s + (0.361 − 0.625i)7-s − 0.269·8-s + (0.403 − 0.699i)9-s + (0.750 + 1.29i)10-s − 1.14·11-s + (−0.259 − 0.449i)12-s + (−0.943 + 1.63i)13-s + (−0.533 + 0.924i)14-s + (−0.223 + 0.386i)15-s − 0.783·16-s + (0.109 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.653 - 0.756i$
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: \((2,\ 43,\ (\ :5/2),\ -0.653 - 0.756i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0157408 + 0.0344044i\)
\(L(\frac12)\) \(\approx\) \(0.0157408 + 0.0344044i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (359. - 1.21e4i)T \)
good2 \( 1 + 8.35T + 32T^{2} \)
3 \( 1 + (3.42 + 5.92i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (28.3 + 49.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-46.8 + 81.1i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 458.T + 1.61e5T^{2} \)
13 \( 1 + (575. - 995. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-130. + 225. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-788. - 1.36e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (825. + 1.43e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (2.56e3 - 4.44e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.87e3 - 4.98e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (4.11e3 + 7.12e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 2.69e3T + 1.15e8T^{2} \)
47 \( 1 + 1.83e4T + 2.29e8T^{2} \)
53 \( 1 + (8.83e3 + 1.53e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 - 1.05e4T + 7.14e8T^{2} \)
61 \( 1 + (1.12e4 - 1.94e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.62e4 + 2.82e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-8.57e3 + 1.48e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (2.64e4 - 4.57e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.06e4 + 1.84e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (5.41e4 + 9.37e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (3.67e3 + 6.36e3i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.36e3T + 8.58e9T^{2} \)
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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

−14.28255410820369158407630643763, −12.67167834060547214895452005769, −11.65493175616592133986816412299, −10.25791002613239068069892806815, −9.153919013288390293168925904225, −7.919602714853518242578387630316, −6.99987590939873791812910465722, −4.59452656659700180820426972166, −1.47288445497945707473858184094, −0.03502963308361126479028959767, 2.55718545102459689832598684671, 5.21049245284755936227147972822, 7.47012271278651530020551532492, 8.036622883446565072062392952776, 9.831422699086784902658065396084, 10.55812033955740957272803357734, 11.51325916928532787228680621965, 13.27749584040412327972119799444, 15.33374589404195041322302590902, 15.54186400296046456238451713877

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