Properties

Degree 2
Conductor 43
Sign $-0.540 + 0.841i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 − 1.38i)2-s + (−5.76 − 0.869i)3-s + (6.42 − 28.1i)4-s + (43.6 + 29.7i)5-s + (5.15 + 8.93i)6-s + (20.6 − 35.7i)7-s + (−96.9 + 46.7i)8-s + (−199. − 61.5i)9-s + (−6.98 − 93.2i)10-s + (−168. − 739. i)11-s + (−61.5 + 156. i)12-s + (11.5 − 153. i)13-s + (−72.2 + 10.8i)14-s + (−226. − 209. i)15-s + (−660. − 318. i)16-s + (920. − 627. i)17-s + ⋯
L(s)  = 1  + (−0.194 − 0.244i)2-s + (−0.370 − 0.0557i)3-s + (0.200 − 0.879i)4-s + (0.781 + 0.532i)5-s + (0.0585 + 0.101i)6-s + (0.159 − 0.275i)7-s + (−0.535 + 0.258i)8-s + (−0.821 − 0.253i)9-s + (−0.0220 − 0.294i)10-s + (−0.420 − 1.84i)11-s + (−0.123 + 0.314i)12-s + (0.0189 − 0.252i)13-s + (−0.0984 + 0.0148i)14-s + (−0.259 − 0.240i)15-s + (−0.645 − 0.310i)16-s + (0.772 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.540 + 0.841i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ -0.540 + 0.841i)\)
\(L(3)\)  \(\approx\)  \(0.555429 - 1.01698i\)
\(L(\frac12)\)  \(\approx\)  \(0.555429 - 1.01698i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-1.06e4 - 5.83e3i)T \)
good2 \( 1 + (1.10 + 1.38i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (5.76 + 0.869i)T + (232. + 71.6i)T^{2} \)
5 \( 1 + (-43.6 - 29.7i)T + (1.14e3 + 2.90e3i)T^{2} \)
7 \( 1 + (-20.6 + 35.7i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (168. + 739. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (-11.5 + 153. i)T + (-3.67e5 - 5.53e4i)T^{2} \)
17 \( 1 + (-920. + 627. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (1.08e3 - 333. i)T + (2.04e6 - 1.39e6i)T^{2} \)
23 \( 1 + (2.05e3 - 1.90e3i)T + (4.80e5 - 6.41e6i)T^{2} \)
29 \( 1 + (-8.02e3 + 1.20e3i)T + (1.95e7 - 6.04e6i)T^{2} \)
31 \( 1 + (-954. + 2.43e3i)T + (-2.09e7 - 1.94e7i)T^{2} \)
37 \( 1 + (-3.34e3 - 5.79e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (2.43e3 + 3.05e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (-5.75e3 + 2.52e4i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (360. + 4.81e3i)T + (-4.13e8 + 6.23e7i)T^{2} \)
59 \( 1 + (4.02e4 + 1.93e4i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (-9.82e3 - 2.50e4i)T + (-6.19e8 + 5.74e8i)T^{2} \)
67 \( 1 + (-2.43e3 + 749. i)T + (1.11e9 - 7.60e8i)T^{2} \)
71 \( 1 + (-4.05e4 - 3.75e4i)T + (1.34e8 + 1.79e9i)T^{2} \)
73 \( 1 + (3.76e3 - 5.02e4i)T + (-2.04e9 - 3.08e8i)T^{2} \)
79 \( 1 + (-4.31e3 + 7.47e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.29e4 - 1.95e3i)T + (3.76e9 + 1.16e9i)T^{2} \)
89 \( 1 + (4.78e4 + 7.20e3i)T + (5.33e9 + 1.64e9i)T^{2} \)
97 \( 1 + (1.05e4 + 4.60e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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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

−14.21622584795551856061304698084, −13.82372904090014895595144175140, −11.77683625819296111955293674169, −10.83299611611946171083231492106, −9.965348052190606961697358499987, −8.409470789118498098569742216761, −6.25790912995454237353404110497, −5.61187044022889740638888513778, −2.79353836464551545377456438315, −0.68430125672089788524182825993, 2.30609503678903951823336360052, 4.71184184089488045515163174367, 6.26740282571047358433491198150, 7.84477350976558160345492987927, 9.051598167629061607324575604224, 10.43355924864909719380748869672, 12.12808272977352921703865904817, 12.66699495204786856854795357534, 14.20407800388739721229387004504, 15.55354470368070301747995235409

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