Properties

Degree 2
Conductor 43
Sign $-0.382 + 0.923i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.4i·2-s + (−30.6 + 17.7i)3-s + 644.·4-s + (−4.83e3 + 2.78e3i)5-s + (344. + 596. i)6-s + (9.59e3 + 5.54e3i)7-s − 3.24e4i·8-s + (−2.88e4 + 5.00e4i)9-s + (5.43e4 + 9.40e4i)10-s − 1.43e5·11-s + (−1.97e4 + 1.14e4i)12-s + (2.30e5 − 3.98e5i)13-s + (1.07e5 − 1.86e5i)14-s + (9.87e4 − 1.71e5i)15-s + 2.79e4·16-s + (1.10e6 − 1.91e6i)17-s + ⋯
L(s)  = 1  − 0.608i·2-s + (−0.126 + 0.0728i)3-s + 0.629·4-s + (−1.54 + 0.892i)5-s + (0.0443 + 0.0767i)6-s + (0.571 + 0.329i)7-s − 0.991i·8-s + (−0.489 + 0.847i)9-s + (0.543 + 0.940i)10-s − 0.890·11-s + (−0.0794 + 0.0458i)12-s + (0.620 − 1.07i)13-s + (0.200 − 0.347i)14-s + (0.130 − 0.225i)15-s + 0.0266·16-s + (0.777 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.382 + 0.923i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.382 + 0.923i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.625869 - 0.936947i\)
\(L(\frac12)\)  \(\approx\)  \(0.625869 - 0.936947i\)
\(L(6)\)   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 \( 1 + (1.30e8 + 6.77e7i)T \)
good2 \( 1 + 19.4iT - 1.02e3T^{2} \)
3 \( 1 + (30.6 - 17.7i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (4.83e3 - 2.78e3i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (-9.59e3 - 5.54e3i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 + 1.43e5T + 2.59e10T^{2} \)
13 \( 1 + (-2.30e5 + 3.98e5i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (-1.10e6 + 1.91e6i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (5.18e4 - 2.99e4i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (-3.32e6 - 5.75e6i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (9.46e6 + 5.46e6i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (2.14e7 + 3.72e7i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (-9.85e7 + 5.69e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 + 1.61e8T + 1.34e16T^{2} \)
47 \( 1 - 9.70e5T + 5.25e16T^{2} \)
53 \( 1 + (-4.49e7 - 7.78e7i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 - 8.85e8T + 5.11e17T^{2} \)
61 \( 1 + (4.64e7 + 2.68e7i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (-9.98e8 - 1.72e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (1.69e9 + 9.78e8i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (-3.11e9 - 1.79e9i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (-1.28e9 + 2.22e9i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (1.99e9 + 3.45e9i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (-3.45e9 + 1.99e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 - 1.75e9T + 7.37e19T^{2} \)
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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

−13.14215183908063457169243233820, −11.54583271183172728245943304284, −11.35041655788943566446183829729, −10.28298084198869887046867742524, −7.993477932809265172923451167142, −7.39626375259174259451327863856, −5.41081739818088867297466821506, −3.49037808039292550820848149287, −2.51949996022980842183160661934, −0.39624418080382889494794917251, 1.26280119759919854977133653935, 3.57366122220483768755269199300, 5.03355117691715117646395704862, 6.62957311058236092281861430117, 7.950951014691969666208086655342, 8.595413801365181412476248947671, 10.91300793150082682016095122651, 11.72413733903673423799130645392, 12.69173836927220007968396116798, 14.58669878677341107920345968979

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