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Properties

Label 2-43-43.37-c4-0-6
Degree $2$
Conductor $43$
Sign $0.902 - 0.430i$
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  − 0.411i·2-s + (10.8 + 6.24i)3-s + 15.8·4-s + (−3.72 − 2.15i)5-s + (2.57 − 4.45i)6-s + (−34.4 + 19.8i)7-s − 13.1i·8-s + (37.5 + 65.0i)9-s + (−0.886 + 1.53i)10-s + 94.7·11-s + (171. + 98.8i)12-s + (−51.3 − 89.0i)13-s + (8.18 + 14.1i)14-s + (−26.8 − 46.5i)15-s + 247.·16-s + (43.0 + 74.6i)17-s + ⋯
L(s)  = 1  − 0.102i·2-s + (1.20 + 0.694i)3-s + 0.989·4-s + (−0.149 − 0.0860i)5-s + (0.0714 − 0.123i)6-s + (−0.702 + 0.405i)7-s − 0.204i·8-s + (0.463 + 0.802i)9-s + (−0.00886 + 0.0153i)10-s + 0.782·11-s + (1.18 + 0.686i)12-s + (−0.304 − 0.526i)13-s + (0.0417 + 0.0723i)14-s + (−0.119 − 0.206i)15-s + 0.968·16-s + (0.149 + 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(4.44490\)
Root analytic conductor: \(2.10829\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :2),\ 0.902 - 0.430i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.26149 + 0.512052i\)
\(L(\frac12)\) \(\approx\) \(2.26149 + 0.512052i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-937. - 1.59e3i)T \)
good2 \( 1 + 0.411iT - 16T^{2} \)
3 \( 1 + (-10.8 - 6.24i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (3.72 + 2.15i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (34.4 - 19.8i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 - 94.7T + 1.46e4T^{2} \)
13 \( 1 + (51.3 + 89.0i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-43.0 - 74.6i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (372. + 214. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (239. - 415. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (662. - 382. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-736. + 1.27e3i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (665. + 384. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 278.T + 2.82e6T^{2} \)
47 \( 1 + 326.T + 4.87e6T^{2} \)
53 \( 1 + (1.21e3 - 2.10e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 - 6.13e3T + 1.21e7T^{2} \)
61 \( 1 + (425. - 245. i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-530. + 918. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (3.89e3 - 2.24e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (996. - 575. i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-217. - 376. i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (2.80e3 - 4.85e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (-1.14e4 - 6.63e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 - 1.52e4T + 8.85e7T^{2} \)
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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

−15.30595530321224052223563037524, −14.48321486001118635690132409103, −12.95771991827115388873406634964, −11.70344353676811788005100605802, −10.22051368995764203822059012513, −9.203119114563539956606977846586, −7.87101824716544597839816098800, −6.23731261183243163133714216554, −3.81793021485388907991648174413, −2.48175580495723107386845692400, 1.95239490603867615112641362105, 3.51093745389100647683969836885, 6.48445275796755857254908652367, 7.36897407833452704077127260084, 8.663346062221730222200394495460, 10.15862162014832920050370966772, 11.75084860504694397193732849405, 12.86313472109712604582110563540, 14.12242051093220045668032820024, 14.89733434124197242147548032103

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