Properties

Degree 2
Conductor 43
Sign $-0.175 - 0.984i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.1i·2-s + (15.3 − 8.86i)3-s − 165.·4-s + (63.9 − 36.9i)5-s + (−134. − 232. i)6-s + (−298. − 172. i)7-s + 1.53e3i·8-s + (−207. + 358. i)9-s + (−559. − 969. i)10-s − 1.17e3·11-s + (−2.54e3 + 1.46e3i)12-s + (642. − 1.11e3i)13-s + (−2.60e3 + 4.51e3i)14-s + (655. − 1.13e3i)15-s + 1.27e4·16-s + (1.58e3 − 2.74e3i)17-s + ⋯
L(s)  = 1  − 1.89i·2-s + (0.568 − 0.328i)3-s − 2.58·4-s + (0.511 − 0.295i)5-s + (−0.622 − 1.07i)6-s + (−0.869 − 0.501i)7-s + 3.00i·8-s + (−0.284 + 0.492i)9-s + (−0.559 − 0.969i)10-s − 0.880·11-s + (−1.47 + 0.849i)12-s + (0.292 − 0.506i)13-s + (−0.950 + 1.64i)14-s + (0.194 − 0.336i)15-s + 3.10·16-s + (0.323 − 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.175 - 0.984i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.175 - 0.984i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.658881 + 0.786493i\)
\(L(\frac12)\)  \(\approx\)  \(0.658881 + 0.786493i\)
\(L(4)\)   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 \( 1 + (-7.91e4 + 7.13e3i)T \)
good2 \( 1 + 15.1iT - 64T^{2} \)
3 \( 1 + (-15.3 + 8.86i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-63.9 + 36.9i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (298. + 172. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 1.17e3T + 1.77e6T^{2} \)
13 \( 1 + (-642. + 1.11e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-1.58e3 + 2.74e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-5.76e3 + 3.33e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (7.35e3 + 1.27e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-3.26e3 - 1.88e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (2.91e4 + 5.05e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (2.24e4 - 1.29e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 2.69e4T + 4.75e9T^{2} \)
47 \( 1 + 1.98e5T + 1.07e10T^{2} \)
53 \( 1 + (1.20e5 + 2.08e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 - 1.63e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.66e5 - 1.53e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.45e5 + 2.52e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-5.00e5 - 2.89e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (-9.58e3 - 5.53e3i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-3.48e5 + 6.04e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-9.46e4 - 1.63e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (-4.23e5 + 2.44e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 8.95e5T + 8.32e11T^{2} \)
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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

−13.19498770571592039644241633947, −13.01185503998995896269093596164, −11.36659944102517265576533321291, −10.21427057927423053581266870416, −9.354674828932611328845388424820, −7.966251035491587050059962415139, −5.24042079799651680655904116058, −3.36517977798371795820027025574, −2.21892293220381868457873886964, −0.44637450956478186238126633273, 3.52063714639869336309684566954, 5.52285971338927144737666415465, 6.45313656296827348355994421742, 7.933260601821586349137050051053, 9.129102817679951965554664993131, 9.923838844983031787020296564451, 12.59353491202717870446752315660, 13.85778259776621702502833432869, 14.49563821687218419054382979669, 15.73715755245099110918048147470

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