Properties

Degree 2
Conductor 43
Sign $-0.0745 + 0.997i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.133i·2-s − 14.8i·3-s + 15.9·4-s + 26.0i·5-s − 1.97·6-s − 87.9i·7-s − 4.26i·8-s − 138.·9-s + 3.47·10-s − 101.·11-s − 236. i·12-s + 190.·13-s − 11.7·14-s + 385.·15-s + 255.·16-s − 227.·17-s + ⋯
L(s)  = 1  − 0.0333i·2-s − 1.64i·3-s + 0.998·4-s + 1.04i·5-s − 0.0548·6-s − 1.79i·7-s − 0.0666i·8-s − 1.70·9-s + 0.0347·10-s − 0.840·11-s − 1.64i·12-s + 1.12·13-s − 0.0598·14-s + 1.71·15-s + 0.996·16-s − 0.788·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0745 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0745 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.0745 + 0.997i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.0745 + 0.997i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.17485 - 1.26590i\)
\(L(\frac12)\)  \(\approx\)  \(1.17485 - 1.26590i\)
\(L(3)\)   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 \( 1 + (137. - 1.84e3i)T \)
good2 \( 1 + 0.133iT - 16T^{2} \)
3 \( 1 + 14.8iT - 81T^{2} \)
5 \( 1 - 26.0iT - 625T^{2} \)
7 \( 1 + 87.9iT - 2.40e3T^{2} \)
11 \( 1 + 101.T + 1.46e4T^{2} \)
13 \( 1 - 190.T + 2.85e4T^{2} \)
17 \( 1 + 227.T + 8.35e4T^{2} \)
19 \( 1 - 304. iT - 1.30e5T^{2} \)
23 \( 1 - 797.T + 2.79e5T^{2} \)
29 \( 1 - 1.08e3iT - 7.07e5T^{2} \)
31 \( 1 - 433.T + 9.23e5T^{2} \)
37 \( 1 + 310. iT - 1.87e6T^{2} \)
41 \( 1 - 1.33e3T + 2.82e6T^{2} \)
47 \( 1 - 1.00e3T + 4.87e6T^{2} \)
53 \( 1 + 3.64e3T + 7.89e6T^{2} \)
59 \( 1 + 1.28e3T + 1.21e7T^{2} \)
61 \( 1 + 2.27e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.75e3T + 2.01e7T^{2} \)
71 \( 1 + 4.25e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.32e3iT - 2.83e7T^{2} \)
79 \( 1 + 227.T + 3.89e7T^{2} \)
83 \( 1 + 1.89e3T + 4.74e7T^{2} \)
89 \( 1 - 1.08e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.09e4T + 8.85e7T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.60420708558352966523831277305, −13.59329147859155880946878761708, −12.74312065542155527304198033460, −11.02484178992664422982967872940, −10.74471567353136241406644807939, −7.891037933038580423995626443439, −7.07465846830812508929073911249, −6.40067771180259219568478316355, −3.06927732898238882247780417775, −1.27544788562586107829633175294, 2.76516038077872148322488566760, 4.86267442932936355387105180397, 5.89833870847558574109162555497, 8.531536419778971310727590575771, 9.223846148068503352117825418777, 10.78457474043951332442525377739, 11.65890310106071817257822446312, 13.00695844548918930409503600616, 15.14143718400814314375438755455, 15.60118151551328094046641428777

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