Properties

Degree 2
Conductor 43
Sign $0.637 - 0.770i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.72 − 7.18i)2-s + (9.19 + 1.38i)3-s + (9.70 − 42.5i)4-s + (445. + 303. i)5-s + (−42.7 − 74.0i)6-s + (−356. + 617. i)7-s + (−1.42e3 + 684. i)8-s + (−2.00e3 − 619. i)9-s + (−370. − 4.94e3i)10-s + (644. + 2.82e3i)11-s + (148. − 377. i)12-s + (−615. + 8.21e3i)13-s + (6.47e3 − 975. i)14-s + (3.68e3 + 3.41e3i)15-s + (8.01e3 + 3.86e3i)16-s + (−2.53e4 + 1.72e4i)17-s + ⋯
L(s)  = 1  + (−0.506 − 0.634i)2-s + (0.196 + 0.0296i)3-s + (0.0758 − 0.332i)4-s + (1.59 + 1.08i)5-s + (−0.0807 − 0.139i)6-s + (−0.392 + 0.680i)7-s + (−0.980 + 0.472i)8-s + (−0.917 − 0.283i)9-s + (−0.117 − 1.56i)10-s + (0.145 + 0.639i)11-s + (0.0247 − 0.0631i)12-s + (−0.0777 + 1.03i)13-s + (0.630 − 0.0950i)14-s + (0.281 + 0.261i)15-s + (0.489 + 0.235i)16-s + (−1.24 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.637 - 0.770i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 0.637 - 0.770i)\)
\(L(4)\)  \(\approx\)  \(1.29311 + 0.608214i\)
\(L(\frac12)\)  \(\approx\)  \(1.29311 + 0.608214i\)
\(L(\frac{9}{2})\)   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 \( 1 + (-4.23e5 - 3.03e5i)T \)
good2 \( 1 + (5.72 + 7.18i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (-9.19 - 1.38i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (-445. - 303. i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (356. - 617. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-644. - 2.82e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (615. - 8.21e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (2.53e4 - 1.72e4i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (-4.50e4 + 1.39e4i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (3.65e4 - 3.39e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (-1.54e5 + 2.33e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (-4.96e4 + 1.26e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (1.57e5 + 2.72e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (1.78e5 + 2.23e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (8.14e4 - 3.56e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (8.14e3 + 1.08e5i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (1.68e5 + 8.10e4i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (2.90e4 + 7.41e4i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (2.08e6 - 6.43e5i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (-2.30e6 - 2.13e6i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (-3.91e5 + 5.23e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (7.28e5 - 1.26e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-3.54e6 - 5.33e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (-3.86e6 - 5.82e5i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (3.21e6 + 1.40e7i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−14.45266700204953299722040630547, −13.70261800017320944471594330646, −11.89768963565768207001222862794, −10.82410646287844729246495152596, −9.592234790964949018140887876015, −9.151457520084599536048267131269, −6.58319087463693344885502609784, −5.74293160852583294507654111892, −2.75924868195032071512033809477, −1.91091028350639436466516533084, 0.66096524520408913535645798927, 2.85950785172557985218517775987, 5.30551179486536912922190605763, 6.50998468759545473267297143709, 8.224364294077882589981992957276, 9.071746436840378920075017557952, 10.20320463161724136867316554975, 12.12031167671620078534739190075, 13.43966219583436924053445651738, 13.95830599280387121192017687041

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