Properties

Degree 2
Conductor 43
Sign $-0.409 + 0.912i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.6i·2-s + 405. i·3-s + 745.·4-s − 2.50e3i·5-s + 6.76e3·6-s − 1.11e4i·7-s − 2.95e4i·8-s − 1.05e5·9-s − 4.18e4·10-s + 7.68e4·11-s + 3.02e5i·12-s − 6.13e5·13-s − 1.85e5·14-s + 1.01e6·15-s + 2.71e5·16-s − 2.69e6·17-s + ⋯
L(s)  = 1  − 0.521i·2-s + 1.67i·3-s + 0.728·4-s − 0.802i·5-s + 0.870·6-s − 0.661i·7-s − 0.900i·8-s − 1.78·9-s − 0.418·10-s + 0.477·11-s + 1.21i·12-s − 1.65·13-s − 0.344·14-s + 1.34·15-s + 0.258·16-s − 1.89·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.409 + 0.912i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.409 + 0.912i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.629743 - 0.972809i\)
\(L(\frac12)\)  \(\approx\)  \(0.629743 - 0.972809i\)
\(L(6)\)   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 \( 1 + (-6.01e7 + 1.34e8i)T \)
good2 \( 1 + 16.6iT - 1.02e3T^{2} \)
3 \( 1 - 405. iT - 5.90e4T^{2} \)
5 \( 1 + 2.50e3iT - 9.76e6T^{2} \)
7 \( 1 + 1.11e4iT - 2.82e8T^{2} \)
11 \( 1 - 7.68e4T + 2.59e10T^{2} \)
13 \( 1 + 6.13e5T + 1.37e11T^{2} \)
17 \( 1 + 2.69e6T + 2.01e12T^{2} \)
19 \( 1 + 2.92e6iT - 6.13e12T^{2} \)
23 \( 1 - 2.15e6T + 4.14e13T^{2} \)
29 \( 1 + 3.40e7iT - 4.20e14T^{2} \)
31 \( 1 + 3.94e7T + 8.19e14T^{2} \)
37 \( 1 - 1.06e8iT - 4.80e15T^{2} \)
41 \( 1 + 6.32e7T + 1.34e16T^{2} \)
47 \( 1 + 7.95e7T + 5.25e16T^{2} \)
53 \( 1 - 5.81e8T + 1.74e17T^{2} \)
59 \( 1 - 6.19e8T + 5.11e17T^{2} \)
61 \( 1 + 7.79e8iT - 7.13e17T^{2} \)
67 \( 1 + 6.11e8T + 1.82e18T^{2} \)
71 \( 1 + 7.76e8iT - 3.25e18T^{2} \)
73 \( 1 - 2.33e8iT - 4.29e18T^{2} \)
79 \( 1 + 2.30e9T + 9.46e18T^{2} \)
83 \( 1 - 9.15e8T + 1.55e19T^{2} \)
89 \( 1 - 4.73e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.01e10T + 7.37e19T^{2} \)
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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

−13.27556947151011994784275830664, −11.79590691222134619443667741433, −10.88312696392129570406557641701, −9.859997214919649024208786069058, −8.957501409971747860282887994316, −6.94747353903272324255367912553, −4.93784891787546814001778554297, −4.06418995302411805696953069564, −2.45854724755955475173281688251, −0.32949994621304945916876958639, 1.85450405856959863719303261701, 2.63856674118069847145061674294, 5.67119789454148018791768177608, 6.91230860262212842913998476207, 7.24459957621074172686842197913, 8.747198155167312444893675871131, 10.89719104193704380455090811209, 11.96450234259281955029062251214, 12.78643392883178107857674965163, 14.42578624949313680506173265291

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