Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4.75·2-s − 7.02·3-s + 14.5·4-s − 4.22·5-s − 33.3·6-s − 11.1·7-s + 31.2·8-s + 22.2·9-s − 20.0·10-s − 43.0·11-s − 102.·12-s + 62.9·13-s − 53.1·14-s + 29.6·15-s + 31.9·16-s + 136.·17-s + 105.·18-s + 108.·19-s − 61.5·20-s + 78.4·21-s − 204.·22-s − 142.·23-s − 219.·24-s − 107.·25-s + 299.·26-s + 33.0·27-s − 162.·28-s + ⋯
L(s)  = 1  + 1.67·2-s − 1.35·3-s + 1.82·4-s − 0.377·5-s − 2.26·6-s − 0.603·7-s + 1.38·8-s + 0.825·9-s − 0.634·10-s − 1.18·11-s − 2.46·12-s + 1.34·13-s − 1.01·14-s + 0.510·15-s + 0.498·16-s + 1.94·17-s + 1.38·18-s + 1.30·19-s − 0.688·20-s + 0.815·21-s − 1.98·22-s − 1.29·23-s − 1.86·24-s − 0.857·25-s + 2.25·26-s + 0.235·27-s − 1.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1849} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 - 4.75T + 8T^{2} \)
3 \( 1 + 7.02T + 27T^{2} \)
5 \( 1 + 4.22T + 125T^{2} \)
7 \( 1 + 11.1T + 343T^{2} \)
11 \( 1 + 43.0T + 1.33e3T^{2} \)
13 \( 1 - 62.9T + 2.19e3T^{2} \)
17 \( 1 - 136.T + 4.91e3T^{2} \)
19 \( 1 - 108.T + 6.85e3T^{2} \)
23 \( 1 + 142.T + 1.21e4T^{2} \)
29 \( 1 - 149.T + 2.43e4T^{2} \)
31 \( 1 + 29.2T + 2.97e4T^{2} \)
37 \( 1 - 69.6T + 5.06e4T^{2} \)
41 \( 1 - 128.T + 6.89e4T^{2} \)
47 \( 1 + 95.1T + 1.03e5T^{2} \)
53 \( 1 + 416.T + 1.48e5T^{2} \)
59 \( 1 + 644.T + 2.05e5T^{2} \)
61 \( 1 + 228.T + 2.26e5T^{2} \)
67 \( 1 + 877.T + 3.00e5T^{2} \)
71 \( 1 + 258.T + 3.57e5T^{2} \)
73 \( 1 + 396.T + 3.89e5T^{2} \)
79 \( 1 - 916.T + 4.93e5T^{2} \)
83 \( 1 + 66.9T + 5.71e5T^{2} \)
89 \( 1 - 365.T + 7.04e5T^{2} \)
97 \( 1 - 1.01e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.037535320671109162501593550549, −7.46884096659371791892972957060, −6.25348539919099908966122649761, −5.94360045662655197005588265343, −5.35944034843413985641227238486, −4.55038967471476878015227539131, −3.52029499714626119087668187501, −2.98568322640789666892204351944, −1.31294116059895365460984436392, 0, 1.31294116059895365460984436392, 2.98568322640789666892204351944, 3.52029499714626119087668187501, 4.55038967471476878015227539131, 5.35944034843413985641227238486, 5.94360045662655197005588265343, 6.25348539919099908966122649761, 7.46884096659371791892972957060, 8.037535320671109162501593550549

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