Properties

Degree 2
Conductor 43
Sign $0.950 - 0.309i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.77i·2-s + 5.61i·3-s + 0.848·4-s + 2.98i·5-s + 9.97·6-s − 6.83i·7-s − 8.60i·8-s − 22.5·9-s + 5.30·10-s + 6.88·11-s + 4.76i·12-s + 12.4·13-s − 12.1·14-s − 16.7·15-s − 11.8·16-s − 15.8·17-s + ⋯
L(s)  = 1  − 0.887i·2-s + 1.87i·3-s + 0.212·4-s + 0.597i·5-s + 1.66·6-s − 0.975i·7-s − 1.07i·8-s − 2.50·9-s + 0.530·10-s + 0.626·11-s + 0.397i·12-s + 0.953·13-s − 0.866·14-s − 1.11·15-s − 0.742·16-s − 0.930·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.950 - 0.309i$
motivic weight  =  \(2\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :1),\ 0.950 - 0.309i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.16181 + 0.184543i\)
\(L(\frac12)\)  \(\approx\)  \(1.16181 + 0.184543i\)
\(L(2)\)   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 \( 1 + (40.8 - 13.3i)T \)
good2 \( 1 + 1.77iT - 4T^{2} \)
3 \( 1 - 5.61iT - 9T^{2} \)
5 \( 1 - 2.98iT - 25T^{2} \)
7 \( 1 + 6.83iT - 49T^{2} \)
11 \( 1 - 6.88T + 121T^{2} \)
13 \( 1 - 12.4T + 169T^{2} \)
17 \( 1 + 15.8T + 289T^{2} \)
19 \( 1 + 19.2iT - 361T^{2} \)
23 \( 1 + 33.2T + 529T^{2} \)
29 \( 1 - 45.8iT - 841T^{2} \)
31 \( 1 - 14.7T + 961T^{2} \)
37 \( 1 - 13.1iT - 1.36e3T^{2} \)
41 \( 1 + 2.49T + 1.68e3T^{2} \)
47 \( 1 + 10.2T + 2.20e3T^{2} \)
53 \( 1 + 31.2T + 2.80e3T^{2} \)
59 \( 1 - 64.4T + 3.48e3T^{2} \)
61 \( 1 - 78.1iT - 3.72e3T^{2} \)
67 \( 1 + 89.3T + 4.48e3T^{2} \)
71 \( 1 + 35.5iT - 5.04e3T^{2} \)
73 \( 1 + 35.9iT - 5.32e3T^{2} \)
79 \( 1 - 50.1T + 6.24e3T^{2} \)
83 \( 1 + 10.3T + 6.88e3T^{2} \)
89 \( 1 - 13.4iT - 7.92e3T^{2} \)
97 \( 1 - 66.6T + 9.40e3T^{2} \)
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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

−15.85168810011880438118077159237, −14.80782705465221954786018806409, −13.60700975974997545924584883730, −11.57603323291820493496265645323, −10.80442796461627068711101137060, −10.21026301611008542074213708510, −8.942529864243737352898801206085, −6.58497577907870624809685895576, −4.33237981399592215235214907634, −3.27195805147200059555584961471, 1.99211827309154029193629155991, 5.84500601107391566529592982909, 6.51090801861167189275620107680, 8.018269683947631578467838493015, 8.682058964884733326247644908125, 11.52968862377680593887183571154, 12.20279881497289732307453008831, 13.41541629873420923091204867162, 14.42556301940616814595607205917, 15.74905087052447038802531667980

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