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

Degree: \(2\)
Conductor: \(43\)
Sign: $0.637 + 0.770i$
Motivic weight: \(7\)
Character: $\chi_{43} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ 0.637 + 0.770i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.29311 - 0.608214i\)
\(L(\frac12)\) \(\approx\) \(1.29311 - 0.608214i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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