Properties

Degree 2
Conductor 43
Sign $-0.530 - 0.847i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2i·2-s − 141. i·3-s + 151.·4-s + 508. i·5-s − 1.44e3·6-s + 610. i·7-s − 4.16e3i·8-s − 1.33e4·9-s + 5.19e3·10-s − 2.33e4·11-s − 2.13e4i·12-s − 3.82e4·13-s + 6.24e3·14-s + 7.17e4·15-s − 3.80e3·16-s + 8.27e3·17-s + ⋯
L(s)  = 1  − 0.638i·2-s − 1.74i·3-s + 0.591·4-s + 0.813i·5-s − 1.11·6-s + 0.254i·7-s − 1.01i·8-s − 2.03·9-s + 0.519·10-s − 1.59·11-s − 1.03i·12-s − 1.33·13-s + 0.162·14-s + 1.41·15-s − 0.0580·16-s + 0.0990·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.530 - 0.847i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.530 - 0.847i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.423560 + 0.764295i\)
\(L(\frac12)\)  \(\approx\)  \(0.423560 + 0.764295i\)
\(L(5)\)   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 \( 1 + (1.81e6 + 2.89e6i)T \)
good2 \( 1 + 10.2iT - 256T^{2} \)
3 \( 1 + 141. iT - 6.56e3T^{2} \)
5 \( 1 - 508. iT - 3.90e5T^{2} \)
7 \( 1 - 610. iT - 5.76e6T^{2} \)
11 \( 1 + 2.33e4T + 2.14e8T^{2} \)
13 \( 1 + 3.82e4T + 8.15e8T^{2} \)
17 \( 1 - 8.27e3T + 6.97e9T^{2} \)
19 \( 1 - 4.39e4iT - 1.69e10T^{2} \)
23 \( 1 + 1.09e5T + 7.83e10T^{2} \)
29 \( 1 + 1.33e6iT - 5.00e11T^{2} \)
31 \( 1 + 6.24e5T + 8.52e11T^{2} \)
37 \( 1 + 1.28e4iT - 3.51e12T^{2} \)
41 \( 1 - 4.22e6T + 7.98e12T^{2} \)
47 \( 1 - 1.43e6T + 2.38e13T^{2} \)
53 \( 1 + 1.03e7T + 6.22e13T^{2} \)
59 \( 1 - 8.60e6T + 1.46e14T^{2} \)
61 \( 1 + 1.46e7iT - 1.91e14T^{2} \)
67 \( 1 + 9.05e6T + 4.06e14T^{2} \)
71 \( 1 + 1.39e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.77e7iT - 8.06e14T^{2} \)
79 \( 1 + 2.98e7T + 1.51e15T^{2} \)
83 \( 1 + 3.68e7T + 2.25e15T^{2} \)
89 \( 1 + 9.08e7iT - 3.93e15T^{2} \)
97 \( 1 + 4.00e7T + 7.83e15T^{2} \)
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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

−13.02881980784803663479731665164, −12.31654333648049722171807011766, −11.30303635455797897735088707546, −10.11163019203987001656966449554, −7.82842661542077392135806109782, −7.16264203701331887058551454219, −5.87764687332680581063221840703, −2.73377352359813864693730188404, −2.15108728313600920306523138924, −0.29083841340393805562481991794, 2.79824140741035278182921035090, 4.76365698932405792987668634944, 5.43498573743155186660501214486, 7.55593273744680749948792987093, 8.879920570656260279259065005712, 10.17735149273073569412855917748, 11.03336289568711984652611126950, 12.58386785073759647410392147852, 14.40811185408229159597203039582, 15.23333521698763959390543651022

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