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Properties

Label 2-43-43.36-c3-0-6
Degree $2$
Conductor $43$
Sign $-0.719 + 0.694i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.190·2-s + (−0.719 − 1.24i)3-s − 7.96·4-s + (−8.17 − 14.1i)5-s + (0.137 + 0.237i)6-s + (−3.11 + 5.40i)7-s + 3.04·8-s + (12.4 − 21.5i)9-s + (1.56 + 2.70i)10-s − 29.4·11-s + (5.72 + 9.91i)12-s + (−11.1 + 19.2i)13-s + (0.595 − 1.03i)14-s + (−11.7 + 20.3i)15-s + 63.1·16-s + (30.1 − 52.2i)17-s + ⋯
L(s)  = 1  − 0.0674·2-s + (−0.138 − 0.239i)3-s − 0.995·4-s + (−0.731 − 1.26i)5-s + (0.00934 + 0.0161i)6-s + (−0.168 + 0.291i)7-s + 0.134·8-s + (0.461 − 0.799i)9-s + (0.0493 + 0.0854i)10-s − 0.807·11-s + (0.137 + 0.238i)12-s + (−0.237 + 0.410i)13-s + (0.0113 − 0.0196i)14-s + (−0.202 + 0.350i)15-s + 0.986·16-s + (0.430 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.719 + 0.694i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.719 + 0.694i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.220095 - 0.545436i\)
\(L(\frac12)\) \(\approx\) \(0.220095 - 0.545436i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (281. - 6.99i)T \)
good2 \( 1 + 0.190T + 8T^{2} \)
3 \( 1 + (0.719 + 1.24i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (8.17 + 14.1i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (3.11 - 5.40i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + 29.4T + 1.33e3T^{2} \)
13 \( 1 + (11.1 - 19.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-30.1 + 52.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-5.01 - 8.69i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-20.3 - 35.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-97.8 + 169. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (120. + 208. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (121. + 210. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 172.T + 6.89e4T^{2} \)
47 \( 1 - 583.T + 1.03e5T^{2} \)
53 \( 1 + (185. + 320. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 714.T + 2.05e5T^{2} \)
61 \( 1 + (353. - 613. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-188. - 326. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-23.7 + 41.0i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-345. + 598. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (144. - 250. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-416. - 722. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (343. + 595. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−15.17309423310702911879068963865, −13.58345083137912296816578142613, −12.63850510130041725253519146005, −11.87570513667901028527474807971, −9.781958954192305530123597849719, −8.836658107790661247851119269712, −7.61365384488094588735482116612, −5.39742396557601890400193843774, −4.09390273254416764835172891137, −0.51798310809664747427969177588, 3.43789747582840056148812587431, 5.05602255824337703811338029454, 7.16206121762026789479971837417, 8.296468129632626073738531159883, 10.24454615101841504030358743672, 10.67708618644676438332682128169, 12.49039742271392730220330670709, 13.67946595067653595774864196154, 14.75634779687162875325759561980, 15.77726078354004745718154822651

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