Properties

Degree 2
Conductor 43
Sign $-0.676 - 0.736i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.1i·2-s + (30.5 + 17.6i)3-s − 39.1·4-s + (101. + 58.6i)5-s + (−179. + 310. i)6-s + (191. − 110. i)7-s + 252. i·8-s + (258. + 448. i)9-s + (−596. + 1.03e3i)10-s + 1.43e3·11-s + (−1.19e3 − 691. i)12-s + (−661. − 1.14e3i)13-s + (1.12e3 + 1.94e3i)14-s + (2.07e3 + 3.58e3i)15-s − 5.06e3·16-s + (−3.95e3 − 6.85e3i)17-s + ⋯
L(s)  = 1  + 1.26i·2-s + (1.13 + 0.653i)3-s − 0.611·4-s + (0.813 + 0.469i)5-s + (−0.830 + 1.43i)6-s + (0.559 − 0.322i)7-s + 0.493i·8-s + (0.355 + 0.615i)9-s + (−0.596 + 1.03i)10-s + 1.08·11-s + (−0.692 − 0.399i)12-s + (−0.301 − 0.521i)13-s + (0.410 + 0.710i)14-s + (0.614 + 1.06i)15-s − 1.23·16-s + (−0.805 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.676 - 0.736i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.676 - 0.736i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.18686 + 2.70097i\)
\(L(\frac12)\)  \(\approx\)  \(1.18686 + 2.70097i\)
\(L(4)\)   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 \( 1 + (5.37e4 - 5.86e4i)T \)
good2 \( 1 - 10.1iT - 64T^{2} \)
3 \( 1 + (-30.5 - 17.6i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (-101. - 58.6i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-191. + 110. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 - 1.43e3T + 1.77e6T^{2} \)
13 \( 1 + (661. + 1.14e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (3.95e3 + 6.85e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (5.83e3 + 3.36e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (7.71e3 - 1.33e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-1.11e4 + 6.42e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-3.86e3 + 6.69e3i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-1.24e4 - 7.17e3i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 7.82e4T + 4.75e9T^{2} \)
47 \( 1 - 1.04e5T + 1.07e10T^{2} \)
53 \( 1 + (2.81e4 - 4.88e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 8.27e4T + 4.21e10T^{2} \)
61 \( 1 + (-1.33e5 + 7.72e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.36e5 - 2.36e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (3.53e5 - 2.03e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (-2.40e5 + 1.38e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-2.80e5 - 4.86e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-3.83e5 + 6.64e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (1.17e6 + 6.75e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 6.96e5T + 8.32e11T^{2} \)
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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

−15.04297844571704394583071015612, −14.21544304591055984717133066825, −13.70905558528504339148184790478, −11.35889683153661745931198444851, −9.762778839066352482476539243083, −8.800412951376181743104889827431, −7.52254573278402416879083004278, −6.21917899254165925255695571630, −4.49628760206762666419151120063, −2.47619622774298945229419989468, 1.59013634620543436638165611597, 2.20917338032817847398973715148, 4.09127032395633584319388132032, 6.50563182678909439027516574725, 8.451313428439935709155108897538, 9.261940467609750239780856798730, 10.65803024531260248030774642304, 12.11719869314839690177103527782, 12.92936103410638823207905115131, 13.97365196924459909779682523947

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