Properties

Degree 2
Conductor 43
Sign $0.349 - 0.936i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.20i·2-s + (−29.5 − 17.0i)3-s + 25.5·4-s + (−46.5 − 26.8i)5-s + (−105. + 183. i)6-s + (−448. + 259. i)7-s − 555. i·8-s + (218. + 378. i)9-s + (−166. + 288. i)10-s + 668.·11-s + (−754. − 435. i)12-s + (1.45e3 + 2.52e3i)13-s + (1.60e3 + 2.78e3i)14-s + (917. + 1.58e3i)15-s − 1.81e3·16-s + (4.62e3 + 8.00e3i)17-s + ⋯
L(s)  = 1  − 0.775i·2-s + (−1.09 − 0.632i)3-s + 0.398·4-s + (−0.372 − 0.215i)5-s + (−0.490 + 0.849i)6-s + (−1.30 + 0.755i)7-s − 1.08i·8-s + (0.299 + 0.519i)9-s + (−0.166 + 0.288i)10-s + 0.502·11-s + (−0.436 − 0.252i)12-s + (0.664 + 1.15i)13-s + (0.585 + 1.01i)14-s + (0.271 + 0.471i)15-s − 0.442·16-s + (0.940 + 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.349 - 0.936i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.349 - 0.936i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.216780 + 0.150458i\)
\(L(\frac12)\)  \(\approx\)  \(0.216780 + 0.150458i\)
\(L(4)\)   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 \( 1 + (7.66e4 + 2.12e4i)T \)
good2 \( 1 + 6.20iT - 64T^{2} \)
3 \( 1 + (29.5 + 17.0i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (46.5 + 26.8i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (448. - 259. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 - 668.T + 1.77e6T^{2} \)
13 \( 1 + (-1.45e3 - 2.52e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (-4.62e3 - 8.00e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (8.61e3 + 4.97e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-1.26e3 + 2.18e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.89e4 - 1.67e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (1.21e4 - 2.11e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-2.02e4 - 1.16e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 1.15e5T + 4.75e9T^{2} \)
47 \( 1 - 6.17e4T + 1.07e10T^{2} \)
53 \( 1 + (7.85e4 - 1.36e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 1.24e5T + 4.21e10T^{2} \)
61 \( 1 + (-1.42e5 + 8.22e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.53e5 - 2.66e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-1.44e5 + 8.36e4i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (9.21e4 - 5.32e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (4.26e4 + 7.38e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-2.15e5 + 3.73e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-3.36e5 - 1.94e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 1.25e6T + 8.32e11T^{2} \)
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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

−15.11598047362730338322164601825, −12.98058230055416195283117022160, −12.36515578804498137261559582158, −11.59999412229275568892861776564, −10.47403147427232446925352799660, −8.943036733313671440407000599440, −6.71649440468010969477700641205, −6.12415058937497287020714225275, −3.65468102228167398611673199722, −1.62527753598556205534325938878, 0.13848413660799481942449066795, 3.54987458622633281369572807999, 5.50302131151754846509552642770, 6.49322617002465495418837983780, 7.71775918588168451124942321848, 9.811488548174890189829300376740, 10.86040163420397750025396287986, 11.82839852610548466449393669243, 13.35974299121166828554029756482, 14.95869449976524323442637853324

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