Properties

Label 2-43-43.12-c6-0-4
Degree $2$
Conductor $43$
Sign $-0.762 - 0.646i$
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  + (0.758 + 1.57i)2-s + (6.18 + 9.07i)3-s + (37.9 − 47.6i)4-s + (−99.6 + 107. i)5-s + (−9.60 + 16.6i)6-s + (−494. + 285. i)7-s + (213. + 48.6i)8-s + (222. − 566. i)9-s + (−244. − 75.5i)10-s + (1.48e3 + 1.85e3i)11-s + (667. + 49.9i)12-s + (−3.55e3 + 1.09e3i)13-s + (−824. − 562. i)14-s + (−1.59e3 − 239. i)15-s + (−782. − 3.42e3i)16-s + (−5.51e3 + 5.11e3i)17-s + ⋯
L(s)  = 1  + (0.0948 + 0.197i)2-s + (0.229 + 0.335i)3-s + (0.593 − 0.744i)4-s + (−0.797 + 0.859i)5-s + (−0.0444 + 0.0769i)6-s + (−1.44 + 0.831i)7-s + (0.416 + 0.0949i)8-s + (0.304 − 0.776i)9-s + (−0.244 − 0.0755i)10-s + (1.11 + 1.39i)11-s + (0.386 + 0.0289i)12-s + (−1.61 + 0.498i)13-s + (−0.300 − 0.204i)14-s + (−0.471 − 0.0710i)15-s + (−0.191 − 0.837i)16-s + (−1.12 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.762 - 0.646i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ -0.762 - 0.646i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.383031 + 1.04404i\)
\(L(\frac12)\) \(\approx\) \(0.383031 + 1.04404i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-6.80e4 - 4.10e4i)T \)
good2 \( 1 + (-0.758 - 1.57i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (-6.18 - 9.07i)T + (-266. + 678. i)T^{2} \)
5 \( 1 + (99.6 - 107. i)T + (-1.16e3 - 1.55e4i)T^{2} \)
7 \( 1 + (494. - 285. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-1.48e3 - 1.85e3i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (3.55e3 - 1.09e3i)T + (3.98e6 - 2.71e6i)T^{2} \)
17 \( 1 + (5.51e3 - 5.11e3i)T + (1.80e6 - 2.40e7i)T^{2} \)
19 \( 1 + (789. - 309. i)T + (3.44e7 - 3.19e7i)T^{2} \)
23 \( 1 + (-789. + 119. i)T + (1.41e8 - 4.36e7i)T^{2} \)
29 \( 1 + (-1.02e4 + 1.50e4i)T + (-2.17e8 - 5.53e8i)T^{2} \)
31 \( 1 + (-11.1 + 149. i)T + (-8.77e8 - 1.32e8i)T^{2} \)
37 \( 1 + (-4.54e4 - 2.62e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (3.55e4 - 1.70e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-8.99e4 + 1.12e5i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (1.21e5 + 3.74e4i)T + (1.83e10 + 1.24e10i)T^{2} \)
59 \( 1 + (-1.20e4 - 5.29e4i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (-1.01e5 + 7.59e3i)T + (5.09e10 - 7.67e9i)T^{2} \)
67 \( 1 + (2.23e4 + 5.69e4i)T + (-6.63e10 + 6.15e10i)T^{2} \)
71 \( 1 + (5.95e4 - 3.94e5i)T + (-1.22e11 - 3.77e10i)T^{2} \)
73 \( 1 + (-1.67e5 - 5.42e5i)T + (-1.25e11 + 8.52e10i)T^{2} \)
79 \( 1 + (-1.71e4 - 2.96e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (1.61e5 - 1.09e5i)T + (1.19e11 - 3.04e11i)T^{2} \)
89 \( 1 + (-1.09e5 - 1.60e5i)T + (-1.81e11 + 4.62e11i)T^{2} \)
97 \( 1 + (1.02e5 + 1.28e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−15.05388641411448588011944866343, −14.71222492899808242878160138994, −12.53631813668173795494677781908, −11.70098447072510614716271060651, −10.03706002413056931293100864532, −9.390822245979497555019436888438, −6.97989195858620798750322737660, −6.49192568007226154643855093895, −4.18693390862446265231392103060, −2.45715611178429754391929221673, 0.46736456096691417960652749547, 2.91288442704404054699071510203, 4.31962293719793786413122226038, 6.78470905982029530360139192341, 7.72465278562442355195140010878, 9.123080749145224451472357639982, 10.82072133946858994891716214423, 12.10972863762675195641960226238, 12.91081465517838041751996247281, 13.86002300861000538973603514979

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