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Properties

Label 2-43-43.36-c5-0-6
Degree $2$
Conductor $43$
Sign $-0.0485 - 0.998i$
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  + 4.50·2-s + (14.9 + 25.9i)3-s − 11.7·4-s + (16.1 + 28.0i)5-s + (67.4 + 116. i)6-s + (70.7 − 122. i)7-s − 196.·8-s + (−328. + 568. i)9-s + (72.9 + 126. i)10-s + 210.·11-s + (−175. − 304. i)12-s + (55.4 − 95.9i)13-s + (318. − 551. i)14-s + (−485. + 841. i)15-s − 510.·16-s + (856. − 1.48e3i)17-s + ⋯
L(s)  = 1  + 0.795·2-s + (0.961 + 1.66i)3-s − 0.366·4-s + (0.289 + 0.501i)5-s + (0.765 + 1.32i)6-s + (0.545 − 0.945i)7-s − 1.08·8-s + (−1.35 + 2.33i)9-s + (0.230 + 0.399i)10-s + 0.523·11-s + (−0.352 − 0.610i)12-s + (0.0909 − 0.157i)13-s + (0.434 − 0.752i)14-s + (−0.557 + 0.965i)15-s − 0.498·16-s + (0.719 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.0485 - 0.998i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ -0.0485 - 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.98666 + 2.08567i\)
\(L(\frac12)\) \(\approx\) \(1.98666 + 2.08567i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (7.75e3 - 9.32e3i)T \)
good2 \( 1 - 4.50T + 32T^{2} \)
3 \( 1 + (-14.9 - 25.9i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-16.1 - 28.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-70.7 + 122. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 210.T + 1.61e5T^{2} \)
13 \( 1 + (-55.4 + 95.9i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-856. + 1.48e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.23e3 - 2.13e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.24e3 - 2.15e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-2.34e3 + 4.05e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (3.51e3 + 6.08e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (3.16e3 + 5.47e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 6.71e3T + 1.15e8T^{2} \)
47 \( 1 + 387.T + 2.29e8T^{2} \)
53 \( 1 + (2.28e3 + 3.96e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + 4.21e3T + 7.14e8T^{2} \)
61 \( 1 + (-7.57e3 + 1.31e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.09e4 + 1.89e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-1.79e4 + 3.10e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (3.39e4 - 5.88e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.76e4 - 4.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (4.75e4 + 8.24e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-1.26e4 - 2.19e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 4.61e4T + 8.58e9T^{2} \)
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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

−14.79203730672558231431737027039, −14.20663900134823509359580306537, −13.65162298141326010119682963476, −11.52205194235796990091801833333, −10.14324041964130795045846883886, −9.403563633576701323563142447056, −7.897640566720123134215596840346, −5.38280477005486386361401693577, −4.16799600753162086170456709911, −3.18625706965043890656821806007, 1.39288196549707791299118236690, 3.10011899883887044273485193025, 5.34980219332260890223230034337, 6.77529690220700872552349356780, 8.570447035655788052040439689722, 8.958804598815656185954595815645, 11.89499263797718136864046668499, 12.58768635115389546197778827139, 13.41304296258383545165105669603, 14.39215669079838377936021157562

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