Properties

Degree 2
Conductor 43
Sign $-0.983 - 0.178i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.806 + 1.67i)2-s + (−16.6 − 1.24i)3-s + (37.7 + 47.3i)4-s + (−23.7 + 76.8i)5-s + (15.4 − 26.8i)6-s + (289. − 167. i)7-s + (−225. + 51.5i)8-s + (−445. − 67.2i)9-s + (−109. − 101. i)10-s + (−36.3 + 45.6i)11-s + (−568. − 834. i)12-s + (−2.47e3 + 2.29e3i)13-s + (46.5 + 620. i)14-s + (489. − 1.24e3i)15-s + (−766. + 3.35e3i)16-s + (−7.10e3 + 2.19e3i)17-s + ⋯
L(s)  = 1  + (−0.100 + 0.209i)2-s + (−0.615 − 0.0461i)3-s + (0.589 + 0.739i)4-s + (−0.189 + 0.614i)5-s + (0.0717 − 0.124i)6-s + (0.845 − 0.488i)7-s + (−0.440 + 0.100i)8-s + (−0.611 − 0.0922i)9-s + (−0.109 − 0.101i)10-s + (−0.0273 + 0.0342i)11-s + (−0.329 − 0.482i)12-s + (−1.12 + 1.04i)13-s + (0.0169 + 0.226i)14-s + (0.145 − 0.369i)15-s + (−0.187 + 0.819i)16-s + (−1.44 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.983 - 0.178i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.983 - 0.178i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.0628490 + 0.697860i\)
\(L(\frac12)\)  \(\approx\)  \(0.0628490 + 0.697860i\)
\(L(4)\)   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 \( 1 + (6.89e3 + 7.92e4i)T \)
good2 \( 1 + (0.806 - 1.67i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (16.6 + 1.24i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (23.7 - 76.8i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (-289. + 167. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (36.3 - 45.6i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (2.47e3 - 2.29e3i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (7.10e3 - 2.19e3i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (1.12e3 + 7.48e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (3.52e3 + 8.97e3i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (2.45e4 - 1.84e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (2.05e4 - 1.39e4i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (-7.55e4 - 4.36e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-4.59e4 - 2.21e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-1.12e5 - 1.41e5i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (1.57e5 + 1.45e5i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (4.63e4 - 2.03e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (8.74e4 - 1.28e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (-1.46e5 + 2.20e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (-6.42e4 - 2.52e4i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (-2.80e5 - 3.02e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (-4.11e4 - 7.12e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (3.24e4 - 4.32e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (3.36e5 + 2.52e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (-4.14e5 + 5.19e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−15.23288555030530989410226345024, −14.31052981067241284648236753466, −12.67321969251985600564579903578, −11.30095572381026837492326156339, −11.07173267420538327453031201849, −8.874618058805254032062614738418, −7.40305881715421809796159839429, −6.50676875664890030574396367410, −4.50043709901076601702704911831, −2.45290061261028171510398874756, 0.33669799728350719996549135407, 2.22602257534275319564112355133, 4.98250532564890690936397854856, 5.90383101815754675038468276289, 7.79013539606566645482380617307, 9.304369263043982991360542938567, 10.79496578097292069204927684262, 11.57183719131817745294423141510, 12.60468966437182122967932692797, 14.42329711660299396227910699506

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