Properties

Label 2-43-43.7-c10-0-32
Degree $2$
Conductor $43$
Sign $0.293 + 0.955i$
Analytic cond. $27.3203$
Root an. cond. $5.22688$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.91i·2-s + (354. − 204. i)3-s + 976.·4-s + (1.40e3 − 813. i)5-s + (−1.41e3 − 2.45e3i)6-s + (5.18e3 + 2.99e3i)7-s − 1.38e4i·8-s + (5.44e4 − 9.43e4i)9-s + (−5.62e3 − 9.74e3i)10-s − 5.04e3·11-s + (3.46e5 − 2.00e5i)12-s + (1.10e5 − 1.90e5i)13-s + (2.06e4 − 3.58e4i)14-s + (3.33e5 − 5.77e5i)15-s + 9.03e5·16-s + (−4.02e5 + 6.97e5i)17-s + ⋯
L(s)  = 1  − 0.216i·2-s + (1.46 − 0.843i)3-s + 0.953·4-s + (0.450 − 0.260i)5-s + (−0.182 − 0.315i)6-s + (0.308 + 0.177i)7-s − 0.422i·8-s + (0.922 − 1.59i)9-s + (−0.0562 − 0.0974i)10-s − 0.0313·11-s + (1.39 − 0.804i)12-s + (0.296 − 0.513i)13-s + (0.0384 − 0.0666i)14-s + (0.439 − 0.760i)15-s + 0.862·16-s + (−0.283 + 0.491i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.293 + 0.955i$
Analytic conductor: \(27.3203\)
Root analytic conductor: \(5.22688\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5),\ 0.293 + 0.955i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(3.65433 - 2.70066i\)
\(L(\frac12)\) \(\approx\) \(3.65433 - 2.70066i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.43e8 - 3.08e7i)T \)
good2 \( 1 + 6.91iT - 1.02e3T^{2} \)
3 \( 1 + (-354. + 204. i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (-1.40e3 + 813. i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (-5.18e3 - 2.99e3i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 + 5.04e3T + 2.59e10T^{2} \)
13 \( 1 + (-1.10e5 + 1.90e5i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (4.02e5 - 6.97e5i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (2.37e6 - 1.36e6i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (-3.41e6 - 5.91e6i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (2.05e7 + 1.18e7i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (-5.43e6 - 9.41e6i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (-1.01e8 + 5.84e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 - 3.32e7T + 1.34e16T^{2} \)
47 \( 1 + 1.35e8T + 5.25e16T^{2} \)
53 \( 1 + (-3.57e7 - 6.19e7i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 + 8.97e8T + 5.11e17T^{2} \)
61 \( 1 + (-4.17e8 - 2.41e8i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (2.16e8 + 3.74e8i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (8.22e8 + 4.74e8i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (-1.16e9 - 6.74e8i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (1.75e9 - 3.04e9i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (-8.25e8 - 1.42e9i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (4.06e9 - 2.34e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 - 8.65e9T + 7.37e19T^{2} \)
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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

−13.32802697299495598493510475962, −12.68489895060350411044879534310, −11.20928514621564674979569412397, −9.664933994689406401233479065986, −8.365299067290757953923481362678, −7.41990807406056938366182437361, −6.03141029903963320615232147379, −3.51037466130556627891159606829, −2.21013903915769906409374286988, −1.41391803035804772330852567018, 1.97170215016588671003186190917, 2.94127509834495850959246185848, 4.51057973931382720943352473975, 6.50841620112020431131425855407, 7.891835491742681244063036595148, 9.028301687563540044682383573291, 10.25483082208386365150926773310, 11.28024804583607618576815954043, 13.19944999794299494448306554432, 14.38055915287342783884808378279

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