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Properties

Label 2-43-43.6-c5-0-3
Degree $2$
Conductor $43$
Sign $0.143 - 0.989i$
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  − 8.55·2-s + (−3.32 + 5.76i)3-s + 41.1·4-s + (8.72 − 15.1i)5-s + (28.4 − 49.2i)6-s + (−112. − 195. i)7-s − 77.9·8-s + (99.3 + 172. i)9-s + (−74.5 + 129. i)10-s − 94.8·11-s + (−136. + 236. i)12-s + (387. + 672. i)13-s + (965. + 1.67e3i)14-s + (58.0 + 100. i)15-s − 649.·16-s + (407. + 705. i)17-s + ⋯
L(s)  = 1  − 1.51·2-s + (−0.213 + 0.369i)3-s + 1.28·4-s + (0.156 − 0.270i)5-s + (0.322 − 0.558i)6-s + (−0.870 − 1.50i)7-s − 0.430·8-s + (0.408 + 0.708i)9-s + (−0.235 + 0.408i)10-s − 0.236·11-s + (−0.274 + 0.474i)12-s + (0.636 + 1.10i)13-s + (1.31 + 2.28i)14-s + (0.0665 + 0.115i)15-s − 0.634·16-s + (0.341 + 0.592i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.143 - 0.989i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ 0.143 - 0.989i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.385692 + 0.333691i\)
\(L(\frac12)\) \(\approx\) \(0.385692 + 0.333691i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-6.82e3 - 1.00e4i)T \)
good2 \( 1 + 8.55T + 32T^{2} \)
3 \( 1 + (3.32 - 5.76i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-8.72 + 15.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (112. + 195. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 94.8T + 1.61e5T^{2} \)
13 \( 1 + (-387. - 672. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-407. - 705. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.25e3 - 2.16e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.28e3 + 2.22e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-1.15e3 - 1.99e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (550. - 952. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.89e3 - 5.02e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 4.81e3T + 1.15e8T^{2} \)
47 \( 1 - 2.85e4T + 2.29e8T^{2} \)
53 \( 1 + (5.55e3 - 9.61e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + 1.23e4T + 7.14e8T^{2} \)
61 \( 1 + (1.83e4 + 3.18e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.01e3 + 3.49e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.89e4 - 3.27e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (1.84e4 + 3.19e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.20e3 - 3.81e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (5.48e3 - 9.50e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (3.01e4 - 5.22e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 5.32e4T + 8.58e9T^{2} \)
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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

−15.94679273742200153623380628589, −14.00505569135308935406912908806, −12.83887069149274003207117573970, −10.76661363982149709445804547544, −10.39779600267898500333658962355, −9.207707337495032251772019439769, −7.81927468108220818639457915211, −6.61504252596519103551094484808, −4.19228313401199291120388836999, −1.32892968936011401996863663624, 0.51801951279039776060760349257, 2.63933250057364262366047640810, 5.92004346741596214451852200712, 7.14643885883743213711629063899, 8.717356692945178914138548771647, 9.497905862157450130979741587123, 10.75710003589557320993794136402, 12.12512742995587674806198984496, 13.20336632268076286328960366372, 15.38163787862803748752456274403

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