Properties

Label 2-43-43.6-c5-0-11
Degree $2$
Conductor $43$
Sign $0.524 - 0.851i$
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  + 9.96·2-s + (−9.83 + 17.0i)3-s + 67.3·4-s + (6.65 − 11.5i)5-s + (−98.0 + 169. i)6-s + (61.0 + 105. i)7-s + 352.·8-s + (−71.8 − 124. i)9-s + (66.3 − 114. i)10-s − 64.2·11-s + (−662. + 1.14e3i)12-s + (38.2 + 66.1i)13-s + (608. + 1.05e3i)14-s + (130. + 226. i)15-s + 1.35e3·16-s + (−984. − 1.70e3i)17-s + ⋯
L(s)  = 1  + 1.76·2-s + (−0.630 + 1.09i)3-s + 2.10·4-s + (0.119 − 0.206i)5-s + (−1.11 + 1.92i)6-s + (0.470 + 0.815i)7-s + 1.94·8-s + (−0.295 − 0.512i)9-s + (0.209 − 0.363i)10-s − 0.159·11-s + (−1.32 + 2.29i)12-s + (0.0627 + 0.108i)13-s + (0.829 + 1.43i)14-s + (0.150 + 0.260i)15-s + 1.32·16-s + (−0.825 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.524 - 0.851i$
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: \((2,\ 43,\ (\ :5/2),\ 0.524 - 0.851i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.19191 + 1.78158i\)
\(L(\frac12)\) \(\approx\) \(3.19191 + 1.78158i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-2.28e3 - 1.19e4i)T \)
good2 \( 1 - 9.96T + 32T^{2} \)
3 \( 1 + (9.83 - 17.0i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-6.65 + 11.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-61.0 - 105. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 64.2T + 1.61e5T^{2} \)
13 \( 1 + (-38.2 - 66.1i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (984. + 1.70e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.24e3 + 2.15e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.29e3 + 3.97e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-887. - 1.53e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (721. - 1.24e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.28e3 - 7.41e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 4.83e3T + 1.15e8T^{2} \)
47 \( 1 + 9.50e3T + 2.29e8T^{2} \)
53 \( 1 + (1.02e4 - 1.77e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + 1.23e4T + 7.14e8T^{2} \)
61 \( 1 + (9.71e3 + 1.68e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.42e4 + 5.93e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-2.17e4 - 3.76e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-2.74e4 - 4.74e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.42e4 + 2.47e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (3.05e3 - 5.29e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (3.71e4 - 6.42e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 6.58e4T + 8.58e9T^{2} \)
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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

−15.25356763495356678483490422555, −14.01261707110960304858588711469, −12.85251411093755660163144930155, −11.57623510745468421283860165739, −10.97852163116473352659144683540, −9.172520851291382547891807907149, −6.75218606536117771741494550931, −5.04032606410674356346924377473, −4.86512682116329196285544892237, −2.83169590593425210603106492518, 1.65192217326734522796925418638, 3.81109786820895079167602922994, 5.49449389775189661171836316468, 6.58170512834885446835571330137, 7.65200107742957051338031572283, 10.63969800360403131257908608928, 11.65132780367982573351414597650, 12.65817737954645890748793150792, 13.44965483850288748248332052973, 14.35207983797527134825912415527

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