Properties

Degree 2
Conductor 43
Sign $0.542 + 0.839i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.84 + 6.08i)2-s + (1.26 + 1.59i)3-s + (−6.33 − 27.7i)4-s + (3.40 + 1.64i)5-s − 15.8·6-s − 175.·7-s + (−24.6 − 11.8i)8-s + (53.1 − 232. i)9-s + (−26.4 + 12.7i)10-s + (62.3 − 273. i)11-s + (36.1 − 45.2i)12-s + (881. + 424. i)13-s + (850. − 1.06e3i)14-s + (1.71 + 7.49i)15-s + (1.01e3 − 487. i)16-s + (−1.21e3 + 586. i)17-s + ⋯
L(s)  = 1  + (−0.857 + 1.07i)2-s + (0.0813 + 0.102i)3-s + (−0.198 − 0.867i)4-s + (0.0609 + 0.0293i)5-s − 0.179·6-s − 1.35·7-s + (−0.135 − 0.0654i)8-s + (0.218 − 0.958i)9-s + (−0.0837 + 0.0403i)10-s + (0.155 − 0.680i)11-s + (0.0724 − 0.0908i)12-s + (1.44 + 0.696i)13-s + (1.16 − 1.45i)14-s + (0.00196 + 0.00860i)15-s + (0.988 − 0.476i)16-s + (−1.02 + 0.492i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.542 + 0.839i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 0.542 + 0.839i)\)
\(L(3)\)  \(\approx\)  \(0.365979 - 0.199273i\)
\(L(\frac12)\)  \(\approx\)  \(0.365979 - 0.199273i\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + (4.96e3 + 1.10e4i)T \)
good2 \( 1 + (4.84 - 6.08i)T + (-7.12 - 31.1i)T^{2} \)
3 \( 1 + (-1.26 - 1.59i)T + (-54.0 + 236. i)T^{2} \)
5 \( 1 + (-3.40 - 1.64i)T + (1.94e3 + 2.44e3i)T^{2} \)
7 \( 1 + 175.T + 1.68e4T^{2} \)
11 \( 1 + (-62.3 + 273. i)T + (-1.45e5 - 6.98e4i)T^{2} \)
13 \( 1 + (-881. - 424. i)T + (2.31e5 + 2.90e5i)T^{2} \)
17 \( 1 + (1.21e3 - 586. i)T + (8.85e5 - 1.11e6i)T^{2} \)
19 \( 1 + (643. + 2.81e3i)T + (-2.23e6 + 1.07e6i)T^{2} \)
23 \( 1 + (-357. + 1.56e3i)T + (-5.79e6 - 2.79e6i)T^{2} \)
29 \( 1 + (3.94e3 - 4.94e3i)T + (-4.56e6 - 1.99e7i)T^{2} \)
31 \( 1 + (469. - 589. i)T + (-6.37e6 - 2.79e7i)T^{2} \)
37 \( 1 + 1.01e4T + 6.93e7T^{2} \)
41 \( 1 + (-6.09e3 + 7.64e3i)T + (-2.57e7 - 1.12e8i)T^{2} \)
47 \( 1 + (-86.7 - 379. i)T + (-2.06e8 + 9.95e7i)T^{2} \)
53 \( 1 + (2.06e4 - 9.96e3i)T + (2.60e8 - 3.26e8i)T^{2} \)
59 \( 1 + (2.38e4 - 1.14e4i)T + (4.45e8 - 5.58e8i)T^{2} \)
61 \( 1 + (9.00e3 + 1.12e4i)T + (-1.87e8 + 8.23e8i)T^{2} \)
67 \( 1 + (8.74e3 + 3.83e4i)T + (-1.21e9 + 5.85e8i)T^{2} \)
71 \( 1 + (1.26e3 + 5.53e3i)T + (-1.62e9 + 7.82e8i)T^{2} \)
73 \( 1 + (-4.92e4 - 2.36e4i)T + (1.29e9 + 1.62e9i)T^{2} \)
79 \( 1 - 5.64e4T + 3.07e9T^{2} \)
83 \( 1 + (-7.62e4 - 9.55e4i)T + (-8.76e8 + 3.84e9i)T^{2} \)
89 \( 1 + (4.03e4 + 5.06e4i)T + (-1.24e9 + 5.44e9i)T^{2} \)
97 \( 1 + (-1.16e4 + 5.09e4i)T + (-7.73e9 - 3.72e9i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.41550838374042585206545963965, −13.76436680330344637757974654073, −12.56410639048434226304814378080, −10.82011172572251627059676723320, −9.163573026912933386112917482233, −8.832489277571736355643814428984, −6.72316202511669188490062666350, −6.32021442759911708000678677246, −3.57562228240295775142407917525, −0.29070209312805336494674431236, 1.78464146834577038132899808656, 3.50628706000120400145781840663, 6.05675898573568043288207023953, 7.931351565552646275008406102713, 9.337882664893645223701588060912, 10.21867285919399764231281902639, 11.26432677414935965223720925294, 12.69696693595966736844383665178, 13.46106004168178834553472386184, 15.36134588641481834516460659629

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