Properties

Label 2-43-43.12-c6-0-18
Degree $2$
Conductor $43$
Sign $-0.940 + 0.339i$
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  + (−1.60 − 3.32i)2-s + (−14.5 − 21.3i)3-s + (31.4 − 39.3i)4-s + (42.5 − 45.8i)5-s + (−47.6 + 82.5i)6-s + (440. − 254. i)7-s + (−411. − 93.9i)8-s + (22.5 − 57.5i)9-s + (−220. − 68.0i)10-s + (409. + 512. i)11-s + (−1.29e3 − 97.2i)12-s + (−2.19e3 + 677. i)13-s + (−1.55e3 − 1.05e3i)14-s + (−1.59e3 − 240. i)15-s + (−370. − 1.62e3i)16-s + (2.21e3 − 2.05e3i)17-s + ⋯
L(s)  = 1  + (−0.200 − 0.415i)2-s + (−0.538 − 0.790i)3-s + (0.490 − 0.615i)4-s + (0.340 − 0.367i)5-s + (−0.220 + 0.382i)6-s + (1.28 − 0.742i)7-s + (−0.803 − 0.183i)8-s + (0.0309 − 0.0789i)9-s + (−0.220 − 0.0680i)10-s + (0.307 + 0.385i)11-s + (−0.750 − 0.0562i)12-s + (−0.999 + 0.308i)13-s + (−0.565 − 0.385i)14-s + (−0.473 − 0.0713i)15-s + (−0.0905 − 0.396i)16-s + (0.450 − 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.940 + 0.339i$
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: \((2,\ 43,\ (\ :3),\ -0.940 + 0.339i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.266328 - 1.52094i\)
\(L(\frac12)\) \(\approx\) \(0.266328 - 1.52094i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-6.95e4 + 3.86e4i)T \)
good2 \( 1 + (1.60 + 3.32i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (14.5 + 21.3i)T + (-266. + 678. i)T^{2} \)
5 \( 1 + (-42.5 + 45.8i)T + (-1.16e3 - 1.55e4i)T^{2} \)
7 \( 1 + (-440. + 254. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-409. - 512. i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (2.19e3 - 677. i)T + (3.98e6 - 2.71e6i)T^{2} \)
17 \( 1 + (-2.21e3 + 2.05e3i)T + (1.80e6 - 2.40e7i)T^{2} \)
19 \( 1 + (9.95e3 - 3.90e3i)T + (3.44e7 - 3.19e7i)T^{2} \)
23 \( 1 + (-1.24e4 + 1.87e3i)T + (1.41e8 - 4.36e7i)T^{2} \)
29 \( 1 + (-1.77e4 + 2.60e4i)T + (-2.17e8 - 5.53e8i)T^{2} \)
31 \( 1 + (1.36e3 - 1.82e4i)T + (-8.77e8 - 1.32e8i)T^{2} \)
37 \( 1 + (-5.44e4 - 3.14e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-3.79e4 + 1.82e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (3.11e4 - 3.91e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (8.90e4 + 2.74e4i)T + (1.83e10 + 1.24e10i)T^{2} \)
59 \( 1 + (5.86e4 + 2.57e5i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (1.82e5 - 1.36e4i)T + (5.09e10 - 7.67e9i)T^{2} \)
67 \( 1 + (1.33e5 + 3.39e5i)T + (-6.63e10 + 6.15e10i)T^{2} \)
71 \( 1 + (-2.61e4 + 1.73e5i)T + (-1.22e11 - 3.77e10i)T^{2} \)
73 \( 1 + (-1.08e5 - 3.51e5i)T + (-1.25e11 + 8.52e10i)T^{2} \)
79 \( 1 + (-1.91e5 - 3.30e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-7.80e5 + 5.32e5i)T + (1.19e11 - 3.04e11i)T^{2} \)
89 \( 1 + (7.71e3 + 1.13e4i)T + (-1.81e11 + 4.62e11i)T^{2} \)
97 \( 1 + (-8.46e5 - 1.06e6i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−14.24641474985156254688661925347, −12.67016031514465293911576494707, −11.74081777022750195229917847391, −10.74135475949954133636474930885, −9.447269366702205349431543341354, −7.57330013340711947893893642041, −6.39008120861509168074292506005, −4.81763606501312919887683150809, −1.90020389337958630659519433207, −0.853536217864648311442238002451, 2.45020466006612870573086075097, 4.67598255342910900256179983432, 6.04979039551822418475856987248, 7.68645153184436545365279295699, 8.909567475181707318005646500478, 10.61129596811636615352566701779, 11.41792231827215431038072700057, 12.62502898525253792994620202648, 14.61498593889363362024676908236, 15.19064993473270801851979650339

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