Properties

Degree 2
Conductor 43
Sign $-0.712 + 0.701i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.95i·2-s + (−14.3 − 8.28i)3-s + 15.6·4-s + (9.86 + 5.69i)5-s + (57.6 − 99.8i)6-s + (−310. + 179. i)7-s + 553. i·8-s + (−227. − 393. i)9-s + (−39.6 + 68.6i)10-s − 1.00e3·11-s + (−224. − 129. i)12-s + (−901. − 1.56e3i)13-s + (−1.24e3 − 2.16e3i)14-s + (−94.4 − 163. i)15-s − 2.85e3·16-s + (−1.33e3 − 2.30e3i)17-s + ⋯
L(s)  = 1  + 0.869i·2-s + (−0.531 − 0.307i)3-s + 0.244·4-s + (0.0789 + 0.0455i)5-s + (0.266 − 0.462i)6-s + (−0.906 + 0.523i)7-s + 1.08i·8-s + (−0.311 − 0.539i)9-s + (−0.0396 + 0.0686i)10-s − 0.755·11-s + (−0.129 − 0.0750i)12-s + (−0.410 − 0.710i)13-s + (−0.454 − 0.787i)14-s + (−0.0279 − 0.0484i)15-s − 0.696·16-s + (−0.271 − 0.469i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.712 + 0.701i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.712 + 0.701i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.0378052 - 0.0923221i\)
\(L(\frac12)\)  \(\approx\)  \(0.0378052 - 0.0923221i\)
\(L(4)\)   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 \( 1 + (-6.04e4 - 5.16e4i)T \)
good2 \( 1 - 6.95iT - 64T^{2} \)
3 \( 1 + (14.3 + 8.28i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (-9.86 - 5.69i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (310. - 179. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + 1.00e3T + 1.77e6T^{2} \)
13 \( 1 + (901. + 1.56e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (1.33e3 + 2.30e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (2.77e3 + 1.60e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (516. - 893. i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.76e4 - 1.59e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (1.25e4 - 2.18e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-1.31e4 - 7.61e3i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 3.44e4T + 4.75e9T^{2} \)
47 \( 1 - 1.01e5T + 1.07e10T^{2} \)
53 \( 1 + (-4.14e4 + 7.18e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 - 1.22e5T + 4.21e10T^{2} \)
61 \( 1 + (3.42e5 - 1.97e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-7.70e4 + 1.33e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (1.72e5 - 9.94e4i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (2.13e4 - 1.23e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (3.06e5 + 5.30e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-2.14e5 + 3.70e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-9.02e5 - 5.21e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 1.53e6T + 8.32e11T^{2} \)
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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

−15.57515970988806275154272879446, −14.59802164816099982708990410791, −13.00320832722987325461769616941, −12.00624286849052721131862475456, −10.69937971232257569857151031678, −9.073530511822047457132675241435, −7.53020524669090961386872919689, −6.34172583816952183849170538527, −5.44936309124028521304391728559, −2.73008944733303785789652300972, 0.04485297255182404206344950036, 2.26624962173278559215462292849, 4.00054675329457957239609053958, 5.91771475739693375486439905021, 7.41777132443605518534604338425, 9.527832371104413389703536729441, 10.53602009363949970899520481058, 11.33838688027347447658302808023, 12.63694855218385344454295396560, 13.56297930575726630291869834615

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