Properties

Label 2-43-43.29-c6-0-9
Degree $2$
Conductor $43$
Sign $0.999 + 0.0155i$
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  + (−0.910 − 1.89i)2-s + (23.8 − 1.78i)3-s + (37.1 − 46.5i)4-s + (47.7 + 154. i)5-s + (−25.1 − 43.4i)6-s + (360. + 208. i)7-s + (−252. − 57.7i)8-s + (−154. + 23.3i)9-s + (249. − 231. i)10-s + (423. + 531. i)11-s + (803. − 1.17e3i)12-s + (−556. − 516. i)13-s + (65.2 − 870. i)14-s + (1.41e3 + 3.61e3i)15-s + (−727. − 3.18e3i)16-s + (6.47e3 + 1.99e3i)17-s + ⋯
L(s)  = 1  + (−0.113 − 0.236i)2-s + (0.883 − 0.0662i)3-s + (0.580 − 0.728i)4-s + (0.382 + 1.23i)5-s + (−0.116 − 0.201i)6-s + (1.05 + 0.606i)7-s + (−0.493 − 0.112i)8-s + (−0.212 + 0.0319i)9-s + (0.249 − 0.231i)10-s + (0.318 + 0.399i)11-s + (0.464 − 0.681i)12-s + (−0.253 − 0.234i)13-s + (0.0237 − 0.317i)14-s + (0.420 + 1.07i)15-s + (−0.177 − 0.778i)16-s + (1.31 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0155i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 + 0.0155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.999 + 0.0155i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ 0.999 + 0.0155i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.69471 - 0.0209957i\)
\(L(\frac12)\) \(\approx\) \(2.69471 - 0.0209957i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-2.21e4 + 7.63e4i)T \)
good2 \( 1 + (0.910 + 1.89i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (-23.8 + 1.78i)T + (720. - 108. i)T^{2} \)
5 \( 1 + (-47.7 - 154. i)T + (-1.29e4 + 8.80e3i)T^{2} \)
7 \( 1 + (-360. - 208. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-423. - 531. i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (556. + 516. i)T + (3.60e5 + 4.81e6i)T^{2} \)
17 \( 1 + (-6.47e3 - 1.99e3i)T + (1.99e7 + 1.35e7i)T^{2} \)
19 \( 1 + (-0.579 + 3.84i)T + (-4.49e7 - 1.38e7i)T^{2} \)
23 \( 1 + (-7.15e3 + 1.82e4i)T + (-1.08e8 - 1.00e8i)T^{2} \)
29 \( 1 + (-8.76e3 - 657. i)T + (5.88e8 + 8.86e7i)T^{2} \)
31 \( 1 + (2.92e4 + 1.99e4i)T + (3.24e8 + 8.26e8i)T^{2} \)
37 \( 1 + (5.52e4 - 3.18e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + (8.70e4 - 4.19e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-6.88e4 + 8.62e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (1.15e5 - 1.07e5i)T + (1.65e9 - 2.21e10i)T^{2} \)
59 \( 1 + (-1.61e4 - 7.07e4i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (-4.58e4 - 6.73e4i)T + (-1.88e10 + 4.79e10i)T^{2} \)
67 \( 1 + (-2.52e5 - 3.80e4i)T + (8.64e10 + 2.66e10i)T^{2} \)
71 \( 1 + (-1.51e5 + 5.96e4i)T + (9.39e10 - 8.71e10i)T^{2} \)
73 \( 1 + (1.87e5 - 2.01e5i)T + (-1.13e10 - 1.50e11i)T^{2} \)
79 \( 1 + (4.15e5 - 7.19e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (6.25e4 + 8.34e5i)T + (-3.23e11 + 4.87e10i)T^{2} \)
89 \( 1 + (8.18e4 - 6.13e3i)T + (4.91e11 - 7.40e10i)T^{2} \)
97 \( 1 + (6.57e5 + 8.23e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−14.70869721744122283543332013275, −14.11052393207543436095933851151, −12.08943253641730283942303753394, −10.92732528855383327750654109561, −9.965423316570469943295162623288, −8.485166635491950420550580561225, −7.02619238595172597326220334427, −5.55349892682987316795705996186, −2.93122336362543680930777504887, −1.88800378598459704145438481885, 1.52993463392597493935139847610, 3.47464677445635701412662238779, 5.30100081018073525779001751565, 7.45588911988770111991618382666, 8.383673310471141355540304943888, 9.320089527024780523212546915200, 11.27727374501224797597206183238, 12.37961044555702248935579198320, 13.70903233225292702876602822978, 14.56022226407430069981608536875

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