Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.86·2-s − 14.8·3-s − 28.5·4-s − 42.2·5-s + 27.6·6-s + 202.·7-s + 112.·8-s − 21.8·9-s + 78.5·10-s + 436.·11-s + 424.·12-s − 617.·13-s − 377.·14-s + 628.·15-s + 703.·16-s + 833.·17-s + 40.6·18-s + 1.35e3·19-s + 1.20e3·20-s − 3.01e3·21-s − 812.·22-s − 904.·23-s − 1.67e3·24-s − 1.34e3·25-s + 1.14e3·26-s + 3.93e3·27-s − 5.79e3·28-s + ⋯
L(s)  = 1  − 0.328·2-s − 0.954·3-s − 0.891·4-s − 0.755·5-s + 0.313·6-s + 1.56·7-s + 0.622·8-s − 0.0898·9-s + 0.248·10-s + 1.08·11-s + 0.850·12-s − 1.01·13-s − 0.514·14-s + 0.720·15-s + 0.687·16-s + 0.699·17-s + 0.0295·18-s + 0.862·19-s + 0.673·20-s − 1.49·21-s − 0.357·22-s − 0.356·23-s − 0.593·24-s − 0.429·25-s + 0.333·26-s + 1.03·27-s − 1.39·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(0.771375\)
\(L(\frac12)\)  \(\approx\)  \(0.771375\)
\(L(\frac{7}{2})\)   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 \( 1 - 1.84e3T \)
good2 \( 1 + 1.86T + 32T^{2} \)
3 \( 1 + 14.8T + 243T^{2} \)
5 \( 1 + 42.2T + 3.12e3T^{2} \)
7 \( 1 - 202.T + 1.68e4T^{2} \)
11 \( 1 - 436.T + 1.61e5T^{2} \)
13 \( 1 + 617.T + 3.71e5T^{2} \)
17 \( 1 - 833.T + 1.41e6T^{2} \)
19 \( 1 - 1.35e3T + 2.47e6T^{2} \)
23 \( 1 + 904.T + 6.43e6T^{2} \)
29 \( 1 - 5.32e3T + 2.05e7T^{2} \)
31 \( 1 - 919.T + 2.86e7T^{2} \)
37 \( 1 - 4.96e3T + 6.93e7T^{2} \)
41 \( 1 + 5.93e3T + 1.15e8T^{2} \)
47 \( 1 - 1.78e4T + 2.29e8T^{2} \)
53 \( 1 - 2.47e4T + 4.18e8T^{2} \)
59 \( 1 - 3.38e4T + 7.14e8T^{2} \)
61 \( 1 - 4.59e4T + 8.44e8T^{2} \)
67 \( 1 + 5.85e4T + 1.35e9T^{2} \)
71 \( 1 - 1.79e3T + 1.80e9T^{2} \)
73 \( 1 + 4.67e4T + 2.07e9T^{2} \)
79 \( 1 + 7.94e4T + 3.07e9T^{2} \)
83 \( 1 - 9.12e4T + 3.93e9T^{2} \)
89 \( 1 + 1.64e4T + 5.58e9T^{2} \)
97 \( 1 - 1.45e5T + 8.58e9T^{2} \)
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

−14.75393250129029423138969816928, −14.04360557292616432918304153379, −12.01930551698968509975794815424, −11.63501157376476368578681355620, −10.16194805356769747799875647556, −8.601699745385936130425124843319, −7.51316870870306800445127543065, −5.38701354559212273686749777793, −4.31147431097332069979659576886, −0.892984695343857725849962469810, 0.892984695343857725849962469810, 4.31147431097332069979659576886, 5.38701354559212273686749777793, 7.51316870870306800445127543065, 8.601699745385936130425124843319, 10.16194805356769747799875647556, 11.63501157376476368578681355620, 12.01930551698968509975794815424, 14.04360557292616432918304153379, 14.75393250129029423138969816928

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