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Properties

Label 2-43-43.6-c3-0-9
Degree $2$
Conductor $43$
Sign $0.868 + 0.496i$
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  + 3.82·2-s + (3.88 − 6.72i)3-s + 6.61·4-s + (−9.06 + 15.7i)5-s + (14.8 − 25.7i)6-s + (−2.59 − 4.50i)7-s − 5.28·8-s + (−16.6 − 28.8i)9-s + (−34.6 + 60.0i)10-s + 31.5·11-s + (25.6 − 44.5i)12-s + (35.6 + 61.7i)13-s + (−9.93 − 17.2i)14-s + (70.4 + 121. i)15-s − 73.1·16-s + (−68.0 − 117. i)17-s + ⋯
L(s)  = 1  + 1.35·2-s + (0.747 − 1.29i)3-s + 0.827·4-s + (−0.810 + 1.40i)5-s + (1.01 − 1.74i)6-s + (−0.140 − 0.243i)7-s − 0.233·8-s + (−0.616 − 1.06i)9-s + (−1.09 + 1.89i)10-s + 0.864·11-s + (0.618 − 1.07i)12-s + (0.760 + 1.31i)13-s + (−0.189 − 0.328i)14-s + (1.21 + 2.09i)15-s − 1.14·16-s + (−0.971 − 1.68i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.868 + 0.496i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ 0.868 + 0.496i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.48222 - 0.659790i\)
\(L(\frac12)\) \(\approx\) \(2.48222 - 0.659790i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-275. + 62.2i)T \)
good2 \( 1 - 3.82T + 8T^{2} \)
3 \( 1 + (-3.88 + 6.72i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (9.06 - 15.7i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (2.59 + 4.50i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 31.5T + 1.33e3T^{2} \)
13 \( 1 + (-35.6 - 61.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (68.0 + 117. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-0.699 + 1.21i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-30.3 + 52.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (14.8 + 25.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (43.6 - 75.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (127. - 221. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 23.1T + 6.89e4T^{2} \)
47 \( 1 + 1.08T + 1.03e5T^{2} \)
53 \( 1 + (-33.4 + 57.9i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 465.T + 2.05e5T^{2} \)
61 \( 1 + (-290. - 502. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (164. - 284. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (416. + 721. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-524. - 908. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (404. + 700. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (519. - 899. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (437. - 757. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 88.7T + 9.12e5T^{2} \)
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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

−14.74849860183495306284876591688, −14.03286514704200353026543401403, −13.46411744448534547365109219854, −11.97629904096880348618421598511, −11.32171240516136770162238806180, −8.878455477073686683537730401988, −6.95289589312207666345963058329, −6.74129228242704513632335057009, −3.99265260313364545083674942650, −2.66373846341936816888733571059, 3.63476446168257839638179675732, 4.30755147756016991728881141995, 5.67191505155204827765318618144, 8.427224573619136822147599422975, 9.156925069474049490126174943549, 10.98748941156216954123097463869, 12.44778138647518303307548004412, 13.18117960808838218909796539790, 14.61991731601488366028294661057, 15.48343949480345981104167991083

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