Properties

Degree 2
Conductor 43
Sign $0.570 - 0.821i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.26 − 19.2i)2-s + (5.00 − 10.3i)3-s + (−124. + 156. i)4-s + (−758. − 173. i)5-s − 246.·6-s − 4.63e3i·7-s + (−1.16e3 − 265. i)8-s + (4.00e3 + 5.02e3i)9-s + (3.69e3 + 1.61e4i)10-s + (−1.47e4 − 1.84e4i)11-s + (1.00e3 + 2.08e3i)12-s + (−5.43e3 + 2.38e4i)13-s + (−8.92e4 + 4.29e4i)14-s + (−5.59e3 + 7.01e3i)15-s + (1.70e4 + 7.47e4i)16-s + (−1.92e4 − 8.43e4i)17-s + ⋯
L(s)  = 1  + (−0.579 − 1.20i)2-s + (0.0618 − 0.128i)3-s + (−0.487 + 0.611i)4-s + (−1.21 − 0.276i)5-s − 0.190·6-s − 1.93i·7-s + (−0.284 − 0.0648i)8-s + (0.610 + 0.765i)9-s + (0.369 + 1.61i)10-s + (−1.00 − 1.25i)11-s + (0.0483 + 0.100i)12-s + (−0.190 + 0.834i)13-s + (−2.32 + 1.11i)14-s + (−0.110 + 0.138i)15-s + (0.260 + 1.14i)16-s + (−0.230 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.570 - 0.821i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.570 - 0.821i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.227150 + 0.118872i\)
\(L(\frac12)\)  \(\approx\)  \(0.227150 + 0.118872i\)
\(L(5)\)   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 \( 1 + (2.68e5 - 3.40e6i)T \)
good2 \( 1 + (9.26 + 19.2i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-5.00 + 10.3i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (758. + 173. i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 + 4.63e3iT - 5.76e6T^{2} \)
11 \( 1 + (1.47e4 + 1.84e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (5.43e3 - 2.38e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (1.92e4 + 8.43e4i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (-8.33e4 - 6.64e4i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-1.10e5 - 1.38e5i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (-2.67e5 - 5.54e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (-1.01e6 + 4.89e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 + 2.37e6iT - 3.51e12T^{2} \)
41 \( 1 + (1.86e6 - 8.98e5i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (2.12e6 - 2.67e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (2.43e6 + 1.06e7i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (-5.93e5 - 2.59e6i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (6.68e6 - 1.38e7i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (4.94e6 - 6.19e6i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (1.63e7 + 1.30e7i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (-2.54e7 - 5.81e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 + 1.25e7T + 1.51e15T^{2} \)
83 \( 1 + (4.90e7 + 2.36e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (-3.81e6 + 7.92e6i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (8.23e7 + 1.03e8i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−13.06819014960526138109164622527, −11.55476589424948066650520007870, −10.91165414678606386574020440658, −9.893695911559794365880598749165, −8.172031821978598411371540549692, −7.24499447937550242851746686766, −4.47722572728818061918659037630, −3.22785606528335282212618213626, −1.13856080446163688988493735684, −0.13658870719989865991824033008, 2.85693657804593761465910991852, 5.02913468761270784514934609928, 6.52854481623835127985776847067, 7.78606515217628818988659625691, 8.645786819262959259904103193691, 9.959804584234920959616488510096, 11.93317227267463607390812805790, 12.54126226594667961671756528061, 15.12838581086061897009798466597, 15.37007277029631880339432008324

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