Properties

Degree 2
Conductor 43
Sign $-0.509 + 0.860i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.95i·2-s + (10.4 + 6.02i)3-s + 60.1·4-s + (−193. − 111. i)5-s + (11.7 − 20.4i)6-s + (−30.2 + 17.4i)7-s − 243. i·8-s + (−291. − 505. i)9-s + (−218. + 378. i)10-s + 688.·11-s + (628. + 362. i)12-s + (−591. − 1.02e3i)13-s + (34.2 + 59.2i)14-s + (−1.34e3 − 2.32e3i)15-s + 3.37e3·16-s + (−2.37e3 − 4.11e3i)17-s + ⋯
L(s)  = 1  − 0.244i·2-s + (0.386 + 0.223i)3-s + 0.940·4-s + (−1.54 − 0.892i)5-s + (0.0546 − 0.0946i)6-s + (−0.0882 + 0.0509i)7-s − 0.474i·8-s + (−0.400 − 0.693i)9-s + (−0.218 + 0.378i)10-s + 0.517·11-s + (0.363 + 0.209i)12-s + (−0.269 − 0.466i)13-s + (0.0124 + 0.0215i)14-s + (−0.398 − 0.690i)15-s + 0.824·16-s + (−0.483 − 0.836i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.509 + 0.860i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.509 + 0.860i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.685409 - 1.20177i\)
\(L(\frac12)\)  \(\approx\)  \(0.685409 - 1.20177i\)
\(L(4)\)   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 \( 1 + (-7.16e4 - 3.44e4i)T \)
good2 \( 1 + 1.95iT - 64T^{2} \)
3 \( 1 + (-10.4 - 6.02i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (193. + 111. i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (30.2 - 17.4i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 - 688.T + 1.77e6T^{2} \)
13 \( 1 + (591. + 1.02e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (2.37e3 + 4.11e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (8.83e3 + 5.09e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-1.29e3 + 2.24e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-3.49e4 + 2.02e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (2.79e4 - 4.84e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-4.05e4 - 2.34e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 4.65e4T + 4.75e9T^{2} \)
47 \( 1 + 5.48e4T + 1.07e10T^{2} \)
53 \( 1 + (-3.18e4 + 5.52e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 2.15e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.13e5 + 1.23e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (4.98e4 - 8.62e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-1.12e5 + 6.50e4i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (-4.45e5 + 2.57e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (2.75e5 + 4.77e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-1.06e5 + 1.84e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-3.22e5 - 1.86e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 7.57e5T + 8.32e11T^{2} \)
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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

−14.67872462179677607463024615629, −12.68334590414505625190332984619, −11.93169714999402779943996509002, −10.98781598728903028408740117858, −9.173351285516444468783862534949, −8.071909797950412427555324156235, −6.66814921767725442898055079352, −4.43331507739744604202507632921, −3.01522909152900735835920185872, −0.59965362851439279646405698690, 2.35980141404366075289378950959, 3.91790167181167949598130700613, 6.40811210287485697801777236025, 7.48611508228676597958478506489, 8.374714401772627818766505241003, 10.69783163700483380793924241710, 11.36236165744705397246402229276, 12.53155588247758631213774628195, 14.44782727491241422570272036960, 14.95130928175221289880753761607

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