Properties

Label 2-43-43.12-c6-0-9
Degree $2$
Conductor $43$
Sign $-0.999 + 0.0209i$
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  + (5.69 + 11.8i)2-s + (18.7 + 27.5i)3-s + (−67.6 + 84.8i)4-s + (55.6 − 59.9i)5-s + (−218. + 379. i)6-s + (−321. + 185. i)7-s + (−569. − 130. i)8-s + (−139. + 356. i)9-s + (1.02e3 + 316. i)10-s + (427. + 536. i)11-s + (−3.60e3 − 270. i)12-s + (−378. + 116. i)13-s + (−4.03e3 − 2.74e3i)14-s + (2.69e3 + 406. i)15-s + (−163. − 714. i)16-s + (4.40e3 − 4.08e3i)17-s + ⋯
L(s)  = 1  + (0.712 + 1.47i)2-s + (0.695 + 1.02i)3-s + (−1.05 + 1.32i)4-s + (0.444 − 0.479i)5-s + (−1.01 + 1.75i)6-s + (−0.937 + 0.541i)7-s + (−1.11 − 0.254i)8-s + (−0.191 + 0.489i)9-s + (1.02 + 0.316i)10-s + (0.321 + 0.402i)11-s + (−2.08 − 0.156i)12-s + (−0.172 + 0.0531i)13-s + (−1.46 − 1.00i)14-s + (0.798 + 0.120i)15-s + (−0.0398 − 0.174i)16-s + (0.895 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0209i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 + 0.0209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.999 + 0.0209i$
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: \((2,\ 43,\ (\ :3),\ -0.999 + 0.0209i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0303517 - 2.89951i\)
\(L(\frac12)\) \(\approx\) \(0.0303517 - 2.89951i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-7.82e4 + 1.43e4i)T \)
good2 \( 1 + (-5.69 - 11.8i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (-18.7 - 27.5i)T + (-266. + 678. i)T^{2} \)
5 \( 1 + (-55.6 + 59.9i)T + (-1.16e3 - 1.55e4i)T^{2} \)
7 \( 1 + (321. - 185. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-427. - 536. i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (378. - 116. i)T + (3.98e6 - 2.71e6i)T^{2} \)
17 \( 1 + (-4.40e3 + 4.08e3i)T + (1.80e6 - 2.40e7i)T^{2} \)
19 \( 1 + (-8.55e3 + 3.35e3i)T + (3.44e7 - 3.19e7i)T^{2} \)
23 \( 1 + (1.50e4 - 2.27e3i)T + (1.41e8 - 4.36e7i)T^{2} \)
29 \( 1 + (6.39e3 - 9.38e3i)T + (-2.17e8 - 5.53e8i)T^{2} \)
31 \( 1 + (-1.20e3 + 1.60e4i)T + (-8.77e8 - 1.32e8i)T^{2} \)
37 \( 1 + (-5.34e4 - 3.08e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (1.12e4 - 5.42e3i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-4.64e4 + 5.81e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (7.10e4 + 2.19e4i)T + (1.83e10 + 1.24e10i)T^{2} \)
59 \( 1 + (2.72e4 + 1.19e5i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (3.44e5 - 2.58e4i)T + (5.09e10 - 7.67e9i)T^{2} \)
67 \( 1 + (-5.45e4 - 1.39e5i)T + (-6.63e10 + 6.15e10i)T^{2} \)
71 \( 1 + (-6.83e4 + 4.53e5i)T + (-1.22e11 - 3.77e10i)T^{2} \)
73 \( 1 + (1.63e5 + 5.29e5i)T + (-1.25e11 + 8.52e10i)T^{2} \)
79 \( 1 + (-2.01e5 - 3.49e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-7.78e5 + 5.30e5i)T + (1.19e11 - 3.04e11i)T^{2} \)
89 \( 1 + (3.70e5 + 5.42e5i)T + (-1.81e11 + 4.62e11i)T^{2} \)
97 \( 1 + (8.73e5 + 1.09e6i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−15.38676001125786578216072233232, −14.37221228347037619753223189607, −13.48889658358786756805159879459, −12.22203227520007066916684717917, −9.716475415726740852594461796441, −9.154288878681770840294375113405, −7.55405072325432441998907091406, −5.98475010329857926660053962129, −4.80042196699344149262073307822, −3.33416064865442311797582891586, 1.14349585099073357837135091524, 2.57295773692358268239954903105, 3.70754453357590826568901195605, 6.07487321379598828202138988255, 7.69138651222546150716886719618, 9.645406596906268275839913402508, 10.54058703655797852111502480767, 12.10127090614612624738044152522, 12.87228655692527688560368470112, 13.93454727728528512860632994393

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