Properties

Degree 2
Conductor 43
Sign $0.0341 + 0.999i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.5i·2-s + (25.2 − 14.5i)3-s − 177.·4-s + (−105. + 61.0i)5-s + (226. + 393. i)6-s + (−10.5 − 6.07i)7-s − 1.77e3i·8-s + (61.0 − 105. i)9-s + (−949. − 1.64e3i)10-s − 1.76e3·11-s + (−4.49e3 + 2.59e3i)12-s + (1.47e3 − 2.55e3i)13-s + (94.4 − 163. i)14-s + (−1.78e3 + 3.08e3i)15-s + 1.61e4·16-s + (397. − 689. i)17-s + ⋯
L(s)  = 1  + 1.94i·2-s + (0.935 − 0.540i)3-s − 2.78·4-s + (−0.845 + 0.488i)5-s + (1.05 + 1.81i)6-s + (−0.0306 − 0.0177i)7-s − 3.46i·8-s + (0.0837 − 0.144i)9-s + (−0.949 − 1.64i)10-s − 1.32·11-s + (−2.60 + 1.50i)12-s + (0.671 − 1.16i)13-s + (0.0344 − 0.0596i)14-s + (−0.527 + 0.913i)15-s + 3.95·16-s + (0.0809 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.0341 + 0.999i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.0341 + 0.999i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.253949 - 0.245414i\)
\(L(\frac12)\)  \(\approx\)  \(0.253949 - 0.245414i\)
\(L(4)\)   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 \( 1 + (7.93e4 + 4.13e3i)T \)
good2 \( 1 - 15.5iT - 64T^{2} \)
3 \( 1 + (-25.2 + 14.5i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (105. - 61.0i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (10.5 + 6.07i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 1.76e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.47e3 + 2.55e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-397. + 689. i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (1.06e4 - 6.13e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-2.84e3 - 4.92e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-531. - 306. i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (3.71e3 + 6.43e3i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (8.67e4 - 5.00e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 2.85e4T + 4.75e9T^{2} \)
47 \( 1 - 6.53e3T + 1.07e10T^{2} \)
53 \( 1 + (-1.95e3 - 3.37e3i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 - 3.51e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.05e5 - 1.18e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-2.02e4 - 3.51e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (2.64e5 + 1.52e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (1.37e5 + 7.92e4i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (1.05e5 - 1.82e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-4.75e5 - 8.22e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (-5.58e5 + 3.22e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.20e6T + 8.32e11T^{2} \)
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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

−15.31065225150218807919657361117, −14.83675584283037260692884728069, −13.54185432929600222167459575230, −12.89111729684651581603764092555, −10.36020998837000991920648449021, −8.397597980588770584857484022097, −8.062960892167784214244638473189, −7.00071080590927172928035952729, −5.41823854488809662225413413938, −3.51376556126503208884351626061, 0.14678293376653745041513948080, 2.29090438842916028662009437939, 3.68400902349200347745133217541, 4.64430664016920665935045490757, 8.404878508861011715803402391460, 8.922549033401082592923060423205, 10.30645658270386146186758870454, 11.31353833338047674710844150326, 12.51163548196310644091713417347, 13.44821459609410795428234941797

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