Properties

Degree 2
Conductor 43
Sign $0.753 + 0.657i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.77 − 12.2i)2-s + (−129. − 19.5i)3-s + (59.2 − 259. i)4-s + (828. + 565. i)5-s + (1.03e3 + 1.78e3i)6-s + (3.21e3 − 5.56e3i)7-s + (−1.09e4 + 5.29e3i)8-s + (−2.30e3 − 710. i)9-s + (−1.17e3 − 1.56e4i)10-s + (1.96e4 + 8.62e4i)11-s + (−1.27e4 + 3.25e4i)12-s + (−1.03e4 + 1.38e5i)13-s + (−9.95e4 + 1.50e4i)14-s + (−9.66e4 − 8.96e4i)15-s + (4.94e4 + 2.38e4i)16-s + (4.33e5 − 2.95e5i)17-s + ⋯
L(s)  = 1  + (−0.431 − 0.541i)2-s + (−0.926 − 0.139i)3-s + (0.115 − 0.506i)4-s + (0.593 + 0.404i)5-s + (0.324 + 0.562i)6-s + (0.505 − 0.875i)7-s + (−0.948 + 0.456i)8-s + (−0.117 − 0.0360i)9-s + (−0.0371 − 0.495i)10-s + (0.405 + 1.77i)11-s + (−0.177 + 0.453i)12-s + (−0.100 + 1.34i)13-s + (−0.692 + 0.104i)14-s + (−0.492 − 0.457i)15-s + (0.188 + 0.0909i)16-s + (1.26 − 0.859i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.753 + 0.657i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ 0.753 + 0.657i)\)
\(L(5)\)  \(\approx\)  \(1.12591 - 0.422024i\)
\(L(\frac12)\)  \(\approx\)  \(1.12591 - 0.422024i\)
\(L(\frac{11}{2})\)   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 \( 1 + (-1.35e7 + 1.78e7i)T \)
good2 \( 1 + (9.77 + 12.2i)T + (-113. + 499. i)T^{2} \)
3 \( 1 + (129. + 19.5i)T + (1.88e4 + 5.80e3i)T^{2} \)
5 \( 1 + (-828. - 565. i)T + (7.13e5 + 1.81e6i)T^{2} \)
7 \( 1 + (-3.21e3 + 5.56e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (-1.96e4 - 8.62e4i)T + (-2.12e9 + 1.02e9i)T^{2} \)
13 \( 1 + (1.03e4 - 1.38e5i)T + (-1.04e10 - 1.58e9i)T^{2} \)
17 \( 1 + (-4.33e5 + 2.95e5i)T + (4.33e10 - 1.10e11i)T^{2} \)
19 \( 1 + (-2.54e4 + 7.85e3i)T + (2.66e11 - 1.81e11i)T^{2} \)
23 \( 1 + (4.62e5 - 4.29e5i)T + (1.34e11 - 1.79e12i)T^{2} \)
29 \( 1 + (-1.71e6 + 2.57e5i)T + (1.38e13 - 4.27e12i)T^{2} \)
31 \( 1 + (-2.23e6 + 5.69e6i)T + (-1.93e13 - 1.79e13i)T^{2} \)
37 \( 1 + (2.27e6 + 3.94e6i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + (-2.08e7 - 2.61e7i)T + (-7.28e13 + 3.19e14i)T^{2} \)
47 \( 1 + (1.94e6 - 8.51e6i)T + (-1.00e15 - 4.85e14i)T^{2} \)
53 \( 1 + (5.89e6 + 7.87e7i)T + (-3.26e15 + 4.91e14i)T^{2} \)
59 \( 1 + (-5.97e7 - 2.87e7i)T + (5.40e15 + 6.77e15i)T^{2} \)
61 \( 1 + (-2.34e7 - 5.96e7i)T + (-8.57e15 + 7.95e15i)T^{2} \)
67 \( 1 + (6.19e7 - 1.91e7i)T + (2.24e16 - 1.53e16i)T^{2} \)
71 \( 1 + (2.50e8 + 2.32e8i)T + (3.42e15 + 4.57e16i)T^{2} \)
73 \( 1 + (1.67e7 - 2.23e8i)T + (-5.82e16 - 8.77e15i)T^{2} \)
79 \( 1 + (-1.20e8 + 2.08e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (-6.49e8 - 9.78e7i)T + (1.78e17 + 5.51e16i)T^{2} \)
89 \( 1 + (5.77e8 + 8.70e7i)T + (3.34e17 + 1.03e17i)T^{2} \)
97 \( 1 + (-2.59e8 - 1.13e9i)T + (-6.84e17 + 3.29e17i)T^{2} \)
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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

−14.07217092235151590499752161080, −12.08913635243122834872168633770, −11.44141174821982640535561077721, −10.19667930841677889454708305106, −9.518225894097677951424378926467, −7.24413605892558866084747301897, −6.10821728680946196384563782722, −4.64293943775337957280312154923, −2.13342128949981918308883051519, −0.920947097896148655957722634643, 0.76983350508033865276136776541, 3.11269618676272690065734035208, 5.56236421694323506048294908683, 5.98870607562439996958263780508, 8.079234118274288475494529265374, 8.829094131674374703784379606247, 10.55392288120939553887137207028, 11.79354038320502873017464241911, 12.64676872612764470074649265694, 14.23384423140329315112304070200

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