Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.31·2-s − 7.10·3-s + 2.98·4-s + 8.51·5-s + 23.5·6-s + 9.84·7-s + 16.6·8-s + 23.4·9-s − 28.2·10-s + 0.597·11-s − 21.2·12-s + 34.9·13-s − 32.6·14-s − 60.5·15-s − 78.9·16-s + 91.3·17-s − 77.7·18-s − 24.0·19-s + 25.4·20-s − 69.9·21-s − 1.98·22-s + 127.·23-s − 118.·24-s − 52.4·25-s − 115.·26-s + 25.0·27-s + 29.3·28-s + ⋯
L(s)  = 1  − 1.17·2-s − 1.36·3-s + 0.373·4-s + 0.761·5-s + 1.60·6-s + 0.531·7-s + 0.734·8-s + 0.869·9-s − 0.892·10-s + 0.0163·11-s − 0.510·12-s + 0.744·13-s − 0.622·14-s − 1.04·15-s − 1.23·16-s + 1.30·17-s − 1.01·18-s − 0.290·19-s + 0.284·20-s − 0.726·21-s − 0.0192·22-s + 1.15·23-s − 1.00·24-s − 0.419·25-s − 0.872·26-s + 0.178·27-s + 0.198·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(0.552289\)
\(L(\frac12)\)  \(\approx\)  \(0.552289\)
\(L(\frac{5}{2})\)   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 \( 1 + 43T \)
good2 \( 1 + 3.31T + 8T^{2} \)
3 \( 1 + 7.10T + 27T^{2} \)
5 \( 1 - 8.51T + 125T^{2} \)
7 \( 1 - 9.84T + 343T^{2} \)
11 \( 1 - 0.597T + 1.33e3T^{2} \)
13 \( 1 - 34.9T + 2.19e3T^{2} \)
17 \( 1 - 91.3T + 4.91e3T^{2} \)
19 \( 1 + 24.0T + 6.85e3T^{2} \)
23 \( 1 - 127.T + 1.21e4T^{2} \)
29 \( 1 - 285.T + 2.43e4T^{2} \)
31 \( 1 + 192.T + 2.97e4T^{2} \)
37 \( 1 - 363.T + 5.06e4T^{2} \)
41 \( 1 - 191.T + 6.89e4T^{2} \)
47 \( 1 + 458.T + 1.03e5T^{2} \)
53 \( 1 + 583.T + 1.48e5T^{2} \)
59 \( 1 - 416.T + 2.05e5T^{2} \)
61 \( 1 + 30.9T + 2.26e5T^{2} \)
67 \( 1 - 16.1T + 3.00e5T^{2} \)
71 \( 1 + 219.T + 3.57e5T^{2} \)
73 \( 1 - 1.03e3T + 3.89e5T^{2} \)
79 \( 1 - 445.T + 4.93e5T^{2} \)
83 \( 1 - 298.T + 5.71e5T^{2} \)
89 \( 1 - 846.T + 7.04e5T^{2} \)
97 \( 1 + 259.T + 9.12e5T^{2} \)
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

−16.18029482951591806374371234538, −14.32536331491568879032141790091, −12.92641654582797863343248816646, −11.42633349989367933165246452880, −10.58824249222854177152601334428, −9.518008966316237270164521160893, −8.037654713362921648572062740172, −6.37730194293215393695599291147, −5.02358574249926313807328819262, −1.12975603138289563510018708281, 1.12975603138289563510018708281, 5.02358574249926313807328819262, 6.37730194293215393695599291147, 8.037654713362921648572062740172, 9.518008966316237270164521160893, 10.58824249222854177152601334428, 11.42633349989367933165246452880, 12.92641654582797863343248816646, 14.32536331491568879032141790091, 16.18029482951591806374371234538

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