Properties

Label 2-43-43.19-c6-0-11
Degree $2$
Conductor $43$
Sign $-0.748 - 0.662i$
Analytic cond. $9.89232$
Root an. cond. $3.14520$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.22 + 5.76i)2-s + (7.33 + 48.6i)3-s + (4.77 + 20.9i)4-s + (108. + 158. i)5-s + (−227. + 394. i)6-s + (411. − 237. i)7-s + (170. − 354. i)8-s + (−1.61e3 + 499. i)9-s + (−132. + 1.77e3i)10-s + (61.7 − 270. i)11-s + (−984. + 386. i)12-s + (−159. − 2.13e3i)13-s + (4.34e3 + 654. i)14-s + (−6.93e3 + 6.43e3i)15-s + (4.51e3 − 2.17e3i)16-s + (−5.87e3 − 4.00e3i)17-s + ⋯
L(s)  = 1  + (0.903 + 0.720i)2-s + (0.271 + 1.80i)3-s + (0.0746 + 0.327i)4-s + (0.866 + 1.27i)5-s + (−1.05 + 1.82i)6-s + (1.20 − 0.692i)7-s + (0.333 − 0.691i)8-s + (−2.22 + 0.685i)9-s + (−0.132 + 1.77i)10-s + (0.0464 − 0.203i)11-s + (−0.569 + 0.223i)12-s + (−0.0727 − 0.970i)13-s + (1.58 + 0.238i)14-s + (−2.05 + 1.90i)15-s + (1.10 − 0.530i)16-s + (−1.19 − 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.662i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.748 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.748 - 0.662i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ -0.748 - 0.662i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.23922 + 3.26953i\)
\(L(\frac12)\) \(\approx\) \(1.23922 + 3.26953i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-2.83e4 + 7.42e4i)T \)
good2 \( 1 + (-7.22 - 5.76i)T + (14.2 + 62.3i)T^{2} \)
3 \( 1 + (-7.33 - 48.6i)T + (-696. + 214. i)T^{2} \)
5 \( 1 + (-108. - 158. i)T + (-5.70e3 + 1.45e4i)T^{2} \)
7 \( 1 + (-411. + 237. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-61.7 + 270. i)T + (-1.59e6 - 7.68e5i)T^{2} \)
13 \( 1 + (159. + 2.13e3i)T + (-4.77e6 + 7.19e5i)T^{2} \)
17 \( 1 + (5.87e3 + 4.00e3i)T + (8.81e6 + 2.24e7i)T^{2} \)
19 \( 1 + (-1.74e3 + 5.64e3i)T + (-3.88e7 - 2.65e7i)T^{2} \)
23 \( 1 + (-6.13e3 - 5.69e3i)T + (1.10e7 + 1.47e8i)T^{2} \)
29 \( 1 + (4.21e3 - 2.79e4i)T + (-5.68e8 - 1.75e8i)T^{2} \)
31 \( 1 + (-3.34e3 - 8.51e3i)T + (-6.50e8 + 6.03e8i)T^{2} \)
37 \( 1 + (4.85e4 + 2.80e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-6.16e3 + 7.73e3i)T + (-1.05e9 - 4.63e9i)T^{2} \)
47 \( 1 + (-1.06e4 - 4.67e4i)T + (-9.71e9 + 4.67e9i)T^{2} \)
53 \( 1 + (1.54e4 - 2.05e5i)T + (-2.19e10 - 3.30e9i)T^{2} \)
59 \( 1 + (3.68e5 - 1.77e5i)T + (2.62e10 - 3.29e10i)T^{2} \)
61 \( 1 + (7.69e4 + 3.02e4i)T + (3.77e10 + 3.50e10i)T^{2} \)
67 \( 1 + (6.20e4 + 1.91e4i)T + (7.47e10 + 5.09e10i)T^{2} \)
71 \( 1 + (1.83e3 + 1.98e3i)T + (-9.57e9 + 1.27e11i)T^{2} \)
73 \( 1 + (-2.21e4 + 1.66e3i)T + (1.49e11 - 2.25e10i)T^{2} \)
79 \( 1 + (-1.00e5 - 1.74e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (2.98e5 - 4.50e4i)T + (3.12e11 - 9.63e10i)T^{2} \)
89 \( 1 + (-2.04e5 - 1.35e6i)T + (-4.74e11 + 1.46e11i)T^{2} \)
97 \( 1 + (-2.79e5 + 1.22e6i)T + (-7.50e11 - 3.61e11i)T^{2} \)
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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

−15.11614125590759096457437287465, −14.14312679169372033944597369754, −13.76318975105060716207774111186, −10.89173469007603256693813589700, −10.59789040838207751603385496786, −9.215443796412210887231726957001, −7.20305458807208919148830567473, −5.52053441016048388403935666128, −4.59149076035575924676254042251, −3.09711357082945573738523313860, 1.62112879978280627118916559764, 2.07964045260721384452711934677, 4.76801403506932593896543193842, 6.09553343752862322275780227428, 8.060685678354341559904901242998, 8.851189298160124261337730865621, 11.45407735408827589037335161351, 12.21692050895545920112147790250, 13.00636341785153724969494113761, 13.74034882156868620668252057492

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