Properties

Degree 2
Conductor 43
Sign $-0.816 - 0.577i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.846i·2-s + 54.5i·3-s + 1.02e3·4-s + 4.88e3i·5-s − 46.1·6-s + 3.06e4i·7-s + 1.73e3i·8-s + 5.60e4·9-s − 4.13e3·10-s − 1.79e5·11-s + 5.58e4i·12-s − 3.38e5·13-s − 2.59e4·14-s − 2.66e5·15-s + 1.04e6·16-s + 7.66e5·17-s + ⋯
L(s)  = 1  + 0.0264i·2-s + 0.224i·3-s + 0.999·4-s + 1.56i·5-s − 0.00593·6-s + 1.82i·7-s + 0.0528i·8-s + 0.949·9-s − 0.0413·10-s − 1.11·11-s + 0.224i·12-s − 0.911·13-s − 0.0481·14-s − 0.350·15-s + 0.997·16-s + 0.539·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.816 - 0.577i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.816 - 0.577i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.669353 + 2.10647i\)
\(L(\frac12)\)  \(\approx\)  \(0.669353 + 2.10647i\)
\(L(6)\)   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 \( 1 + (-1.20e8 - 8.48e7i)T \)
good2 \( 1 - 0.846iT - 1.02e3T^{2} \)
3 \( 1 - 54.5iT - 5.90e4T^{2} \)
5 \( 1 - 4.88e3iT - 9.76e6T^{2} \)
7 \( 1 - 3.06e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.79e5T + 2.59e10T^{2} \)
13 \( 1 + 3.38e5T + 1.37e11T^{2} \)
17 \( 1 - 7.66e5T + 2.01e12T^{2} \)
19 \( 1 + 3.89e6iT - 6.13e12T^{2} \)
23 \( 1 - 3.13e6T + 4.14e13T^{2} \)
29 \( 1 + 1.81e7iT - 4.20e14T^{2} \)
31 \( 1 + 2.96e7T + 8.19e14T^{2} \)
37 \( 1 - 8.05e7iT - 4.80e15T^{2} \)
41 \( 1 - 4.52e7T + 1.34e16T^{2} \)
47 \( 1 - 1.36e8T + 5.25e16T^{2} \)
53 \( 1 + 3.81e8T + 1.74e17T^{2} \)
59 \( 1 - 1.06e9T + 5.11e17T^{2} \)
61 \( 1 - 7.22e8iT - 7.13e17T^{2} \)
67 \( 1 - 1.44e9T + 1.82e18T^{2} \)
71 \( 1 - 3.79e8iT - 3.25e18T^{2} \)
73 \( 1 + 1.13e9iT - 4.29e18T^{2} \)
79 \( 1 + 4.58e9T + 9.46e18T^{2} \)
83 \( 1 - 6.19e9T + 1.55e19T^{2} \)
89 \( 1 - 7.46e9iT - 3.11e19T^{2} \)
97 \( 1 + 3.79e9T + 7.37e19T^{2} \)
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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

−14.79717474456796988988003695335, −12.84333375729468729637661207940, −11.67240170813278747671482681367, −10.71886198520683050972350481628, −9.620842160088433836985410614044, −7.67046943064288807532801437297, −6.68043996355485809839068603559, −5.35510915146534396074080038708, −2.84761396207407435378252137709, −2.35137859192915080407905624327, 0.68326096846780782324351477062, 1.68138611818319326287570076046, 3.91154106139746228186690903324, 5.30053178719435587551213756244, 7.24799276050581089478154236092, 7.83290768935709968683112689072, 9.871573875060642626578178346112, 10.71876792522165712005882030296, 12.49434474726854007399732400788, 12.87036057766167688312303336604

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