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Properties

Label 2-43-43.7-c4-0-5
Degree $2$
Conductor $43$
Sign $0.952 - 0.303i$
Analytic cond. $4.44490$
Root an. cond. $2.10829$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.45i·2-s + (−7.53 + 4.34i)3-s + 9.97·4-s + (−12.0 + 6.95i)5-s + (10.6 + 18.4i)6-s + (83.6 + 48.2i)7-s − 63.7i·8-s + (−2.68 + 4.64i)9-s + (17.0 + 29.5i)10-s + 176.·11-s + (−75.1 + 43.3i)12-s + (−73.8 + 127. i)13-s + (118. − 205. i)14-s + (60.4 − 104. i)15-s + 3.21·16-s + (127. − 220. i)17-s + ⋯
L(s)  = 1  − 0.613i·2-s + (−0.836 + 0.483i)3-s + 0.623·4-s + (−0.481 + 0.278i)5-s + (0.296 + 0.513i)6-s + (1.70 + 0.985i)7-s − 0.996i·8-s + (−0.0331 + 0.0573i)9-s + (0.170 + 0.295i)10-s + 1.46·11-s + (−0.521 + 0.301i)12-s + (−0.437 + 0.757i)13-s + (0.604 − 1.04i)14-s + (0.268 − 0.465i)15-s + 0.0125·16-s + (0.440 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.952 - 0.303i$
Analytic conductor: \(4.44490\)
Root analytic conductor: \(2.10829\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :2),\ 0.952 - 0.303i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.43323 + 0.222998i\)
\(L(\frac12)\) \(\approx\) \(1.43323 + 0.222998i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (407. + 1.80e3i)T \)
good2 \( 1 + 2.45iT - 16T^{2} \)
3 \( 1 + (7.53 - 4.34i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (12.0 - 6.95i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-83.6 - 48.2i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 - 176.T + 1.46e4T^{2} \)
13 \( 1 + (73.8 - 127. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-127. + 220. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (409. - 236. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-193. - 335. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (576. + 333. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (270. + 468. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-1.90e3 + 1.09e3i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 2.05e3T + 2.82e6T^{2} \)
47 \( 1 + 825.T + 4.87e6T^{2} \)
53 \( 1 + (1.13e3 + 1.96e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + 3.85e3T + 1.21e7T^{2} \)
61 \( 1 + (631. + 364. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (262. + 454. i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (1.17e3 + 676. i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (4.15e3 + 2.39e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-1.64e3 + 2.85e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (1.29e3 + 2.24e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (7.87e3 - 4.54e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 1.84e3T + 8.85e7T^{2} \)
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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

−15.17444535234683904720436296930, −14.45007994080407133824721063102, −12.13707700188119835577457913915, −11.46172573878097986349718579336, −11.09996982687435487826947053304, −9.353295141988304695686435873900, −7.60267792164054239016174829859, −5.87702030173629878347413546692, −4.27933570333491459611534064928, −1.90569595872733548014685886805, 1.23044034726465121099778405021, 4.53929530798679136504168204296, 6.16350244040950657144886154312, 7.34736322110999356966110823531, 8.376916181552665558604713955086, 10.85651860149004104595344712069, 11.46926490746403831805179520781, 12.52138163077961237643248416882, 14.50191540563920481817138199716, 14.91219708430436709846005247254

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