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Properties

Label 2-43-43.6-c3-0-0
Degree $2$
Conductor $43$
Sign $-0.439 + 0.898i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.31·2-s + (−4.02 + 6.97i)3-s + 20.2·4-s + (−7.95 + 13.7i)5-s + (21.4 − 37.0i)6-s + (−8.31 − 14.4i)7-s − 64.9·8-s + (−18.9 − 32.8i)9-s + (42.2 − 73.2i)10-s + 31.4·11-s + (−81.4 + 141. i)12-s + (−4.55 − 7.88i)13-s + (44.1 + 76.5i)14-s + (−64.1 − 111. i)15-s + 183.·16-s + (−15.2 − 26.3i)17-s + ⋯
L(s)  = 1  − 1.87·2-s + (−0.775 + 1.34i)3-s + 2.52·4-s + (−0.711 + 1.23i)5-s + (1.45 − 2.52i)6-s + (−0.449 − 0.777i)7-s − 2.87·8-s + (−0.701 − 1.21i)9-s + (1.33 − 2.31i)10-s + 0.861·11-s + (−1.96 + 3.39i)12-s + (−0.0971 − 0.168i)13-s + (0.843 + 1.46i)14-s + (−1.10 − 1.91i)15-s + 2.86·16-s + (−0.217 − 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.439 + 0.898i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.439 + 0.898i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0366080 - 0.0586766i\)
\(L(\frac12)\) \(\approx\) \(0.0366080 - 0.0586766i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-79.8 - 270. i)T \)
good2 \( 1 + 5.31T + 8T^{2} \)
3 \( 1 + (4.02 - 6.97i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (7.95 - 13.7i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (8.31 + 14.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 31.4T + 1.33e3T^{2} \)
13 \( 1 + (4.55 + 7.88i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (15.2 + 26.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (3.71 - 6.43i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (27.1 - 47.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (132. + 229. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (157. - 272. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-103. + 178. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 174.T + 6.89e4T^{2} \)
47 \( 1 + 356.T + 1.03e5T^{2} \)
53 \( 1 + (-84.9 + 147. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 673.T + 2.05e5T^{2} \)
61 \( 1 + (80.9 + 140. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (59.1 - 102. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-45.0 - 78.0i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-49.0 - 84.9i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (232. + 402. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-74.9 + 129. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (399. - 692. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 862.T + 9.12e5T^{2} \)
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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

−16.42653824824642217426420917386, −15.65576494910262703507968200602, −14.71027396981081267681900407046, −11.66332465405469877460472548913, −11.00805063495089252177332024677, −10.20931597316758621241133892121, −9.337573600861445583167896273821, −7.52149565810856526385982181676, −6.43319433528639911404913831016, −3.58075733377641212559327613120, 0.11057647636907200935683790711, 1.59493743261328366571313292256, 6.03848382131296559419665895620, 7.24479419579752906143505450598, 8.420706611596508137881291336270, 9.306495566282822031121412435227, 11.23261278450480873208452520766, 12.06077952932813592066727898947, 12.74919727049695368615567557726, 15.31842232068150323627151202539

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