Properties

Degree 2
Conductor 43
Sign $-0.559 - 0.829i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.05 − 2.19i)2-s + (−37.2 − 2.79i)3-s + (36.2 + 45.3i)4-s + (67.3 − 218. i)5-s + (−45.5 + 78.9i)6-s + (−230. + 133. i)7-s + (289. − 66.1i)8-s + (661. + 99.7i)9-s + (−408. − 378. i)10-s + (−1.42e3 + 1.79e3i)11-s + (−1.22e3 − 1.79e3i)12-s + (−825. + 766. i)13-s + (48.5 + 647. i)14-s + (−3.12e3 + 7.95e3i)15-s + (−665. + 2.91e3i)16-s + (−3.04e3 + 938. i)17-s + ⋯
L(s)  = 1  + (0.132 − 0.274i)2-s + (−1.38 − 0.103i)3-s + (0.565 + 0.709i)4-s + (0.538 − 1.74i)5-s + (−0.210 + 0.365i)6-s + (−0.672 + 0.388i)7-s + (0.566 − 0.129i)8-s + (0.907 + 0.136i)9-s + (−0.408 − 0.378i)10-s + (−1.07 + 1.34i)11-s + (−0.707 − 1.03i)12-s + (−0.375 + 0.348i)13-s + (0.0176 + 0.235i)14-s + (−0.925 + 2.35i)15-s + (−0.162 + 0.711i)16-s + (−0.619 + 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.559 - 0.829i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.559 - 0.829i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.151679 + 0.285242i\)
\(L(\frac12)\)  \(\approx\)  \(0.151679 + 0.285242i\)
\(L(4)\)   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 \( 1 + (-5.18e4 + 6.02e4i)T \)
good2 \( 1 + (-1.05 + 2.19i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (37.2 + 2.79i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (-67.3 + 218. i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (230. - 133. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (1.42e3 - 1.79e3i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (825. - 766. i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (3.04e3 - 938. i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (-192. - 1.27e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (-2.88e3 - 7.35e3i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (7.70e3 - 577. i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (2.20e4 - 1.50e4i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (8.59e4 + 4.96e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-7.89e4 - 3.79e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (4.69e4 + 5.89e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-8.37e4 - 7.77e4i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (3.40e4 - 1.49e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-1.17e5 + 1.72e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (2.08e5 - 3.14e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (6.00e5 + 2.35e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (3.33e4 + 3.59e4i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (1.57e5 + 2.72e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-2.73e4 + 3.65e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (1.49e5 + 1.12e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (-2.82e5 + 3.53e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−15.74097062229895640518143684820, −13.10700913373675267826605572210, −12.58027093745620828741285473064, −11.96098824460730429212734783997, −10.45349877926329269714883534405, −9.065770217641131358442570993267, −7.28087289379790939534424597162, −5.69744477496480057486483765005, −4.62427151228645555001977962770, −1.88650370004147889349488098569, 0.15806549210700869988338065534, 2.78908560370376567130452419953, 5.49250222685496639378403993788, 6.29333128482684045056424738292, 7.16383615663178237742315953992, 10.11003462941024762634927129823, 10.74964039518030039283110621645, 11.32034500031678663701557995501, 13.29175783329977401897837473452, 14.39941412038845495190073100579

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