Properties

Degree 2
Conductor 43
Sign $-0.999 + 0.0292i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.49 − 5.17i)2-s + (−7.43 − 0.557i)3-s + (−10.5 − 13.2i)4-s + (9.68 − 31.4i)5-s + (−21.4 + 37.0i)6-s + (−66.6 + 38.4i)7-s + (−5.54 + 1.26i)8-s + (−25.1 − 3.78i)9-s + (−138. − 128. i)10-s + (42.0 − 52.7i)11-s + (71.3 + 104. i)12-s + (46.0 − 42.7i)13-s + (33.0 + 440. i)14-s + (−89.5 + 228. i)15-s + (53.2 − 233. i)16-s + (364. − 112. i)17-s + ⋯
L(s)  = 1  + (0.623 − 1.29i)2-s + (−0.826 − 0.0619i)3-s + (−0.662 − 0.830i)4-s + (0.387 − 1.25i)5-s + (−0.594 + 1.03i)6-s + (−1.36 + 0.785i)7-s + (−0.0866 + 0.0197i)8-s + (−0.309 − 0.0467i)9-s + (−1.38 − 1.28i)10-s + (0.347 − 0.435i)11-s + (0.495 + 0.726i)12-s + (0.272 − 0.252i)13-s + (0.168 + 2.24i)14-s + (−0.397 + 1.01i)15-s + (0.207 − 0.910i)16-s + (1.25 − 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0292i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.999 + 0.0292i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.999 + 0.0292i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.0190019 - 1.29799i\)
\(L(\frac12)\)  \(\approx\)  \(0.0190019 - 1.29799i\)
\(L(3)\)   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 \( 1 + (-538. - 1.76e3i)T \)
good2 \( 1 + (-2.49 + 5.17i)T + (-9.97 - 12.5i)T^{2} \)
3 \( 1 + (7.43 + 0.557i)T + (80.0 + 12.0i)T^{2} \)
5 \( 1 + (-9.68 + 31.4i)T + (-516. - 352. i)T^{2} \)
7 \( 1 + (66.6 - 38.4i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-42.0 + 52.7i)T + (-3.25e3 - 1.42e4i)T^{2} \)
13 \( 1 + (-46.0 + 42.7i)T + (2.13e3 - 2.84e4i)T^{2} \)
17 \( 1 + (-364. + 112. i)T + (6.90e4 - 4.70e4i)T^{2} \)
19 \( 1 + (36.5 + 242. i)T + (-1.24e5 + 3.84e4i)T^{2} \)
23 \( 1 + (190. + 485. i)T + (-2.05e5 + 1.90e5i)T^{2} \)
29 \( 1 + (-1.04e3 + 77.9i)T + (6.99e5 - 1.05e5i)T^{2} \)
31 \( 1 + (1.41e3 - 962. i)T + (3.37e5 - 8.59e5i)T^{2} \)
37 \( 1 + (-814. - 470. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (-844. - 406. i)T + (1.76e6 + 2.20e6i)T^{2} \)
47 \( 1 + (217. + 273. i)T + (-1.08e6 + 4.75e6i)T^{2} \)
53 \( 1 + (2.55e3 + 2.36e3i)T + (5.89e5 + 7.86e6i)T^{2} \)
59 \( 1 + (319. - 1.40e3i)T + (-1.09e7 - 5.25e6i)T^{2} \)
61 \( 1 + (1.74e3 - 2.56e3i)T + (-5.05e6 - 1.28e7i)T^{2} \)
67 \( 1 + (-8.01e3 + 1.20e3i)T + (1.92e7 - 5.93e6i)T^{2} \)
71 \( 1 + (-676. - 265. i)T + (1.86e7 + 1.72e7i)T^{2} \)
73 \( 1 + (4.05e3 + 4.37e3i)T + (-2.12e6 + 2.83e7i)T^{2} \)
79 \( 1 + (-3.18e3 - 5.51e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-295. + 3.93e3i)T + (-4.69e7 - 7.07e6i)T^{2} \)
89 \( 1 + (6.41e3 + 480. i)T + (6.20e7 + 9.35e6i)T^{2} \)
97 \( 1 + (-2.03e3 + 2.55e3i)T + (-1.96e7 - 8.63e7i)T^{2} \)
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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

−14.11076885780562581492776942245, −12.71604242462607416739598675251, −12.50518173634325657368662939136, −11.38940390696827163993202154182, −9.978512063016389211888939915646, −8.861244800442694766586365245407, −6.11709979394159934927967796941, −5.02214538461822334310349871714, −3.05599281154660912571905596896, −0.78832088008060179796386586616, 3.68574702006922159380640458372, 5.80230633551832219019862862629, 6.46391518294466381751578857679, 7.49238052313901983318145896919, 9.889554054720626481152246349821, 10.87973008392802210590426939941, 12.55710792567798628453387071174, 13.91858314939893317562956814612, 14.52868476818959816704463372246, 15.87865473932709345227933748905

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