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  − 4.78·2-s + 1.15·3-s + 14.8·4-s − 21.3·5-s − 5.53·6-s + 24.0·7-s − 32.7·8-s − 25.6·9-s + 101.·10-s + 17.5·11-s + 17.1·12-s − 56.9·13-s − 114.·14-s − 24.6·15-s + 37.8·16-s − 74.3·17-s + 122.·18-s + 100.·19-s − 316.·20-s + 27.8·21-s − 84.1·22-s − 13.9·23-s − 37.9·24-s + 328.·25-s + 272.·26-s − 60.9·27-s + 357.·28-s + ⋯
L(s)  = 1  − 1.69·2-s + 0.222·3-s + 1.85·4-s − 1.90·5-s − 0.376·6-s + 1.29·7-s − 1.44·8-s − 0.950·9-s + 3.22·10-s + 0.482·11-s + 0.413·12-s − 1.21·13-s − 2.19·14-s − 0.424·15-s + 0.591·16-s − 1.06·17-s + 1.60·18-s + 1.21·19-s − 3.53·20-s + 0.289·21-s − 0.815·22-s − 0.126·23-s − 0.322·24-s + 2.63·25-s + 2.05·26-s − 0.434·27-s + 2.41·28-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 \( 1 + 4.78T + 8T^{2} \)
3 \( 1 - 1.15T + 27T^{2} \)
5 \( 1 + 21.3T + 125T^{2} \)
7 \( 1 - 24.0T + 343T^{2} \)
11 \( 1 - 17.5T + 1.33e3T^{2} \)
13 \( 1 + 56.9T + 2.19e3T^{2} \)
17 \( 1 + 74.3T + 4.91e3T^{2} \)
19 \( 1 - 100.T + 6.85e3T^{2} \)
23 \( 1 + 13.9T + 1.21e4T^{2} \)
29 \( 1 - 87.1T + 2.43e4T^{2} \)
31 \( 1 + 127.T + 2.97e4T^{2} \)
37 \( 1 - 125.T + 5.06e4T^{2} \)
41 \( 1 + 82.7T + 6.89e4T^{2} \)
47 \( 1 + 71.5T + 1.03e5T^{2} \)
53 \( 1 - 152.T + 1.48e5T^{2} \)
59 \( 1 - 319.T + 2.05e5T^{2} \)
61 \( 1 + 353.T + 2.26e5T^{2} \)
67 \( 1 - 37.2T + 3.00e5T^{2} \)
71 \( 1 + 781.T + 3.57e5T^{2} \)
73 \( 1 - 1.17e3T + 3.89e5T^{2} \)
79 \( 1 - 841.T + 4.93e5T^{2} \)
83 \( 1 + 1.11e3T + 5.71e5T^{2} \)
89 \( 1 - 308.T + 7.04e5T^{2} \)
97 \( 1 - 1.10e3T + 9.12e5T^{2} \)
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

−8.569149204901299047286666635043, −7.69562237583185992827478144333, −7.56881010058525996599806492907, −6.66717941053630213076782229667, −5.11882696749182055261433484879, −4.33484362431348564048155674336, −3.14409382864127159893868309533, −2.12948184992125573446259980765, −0.864469026376641852906529146228, 0, 0.864469026376641852906529146228, 2.12948184992125573446259980765, 3.14409382864127159893868309533, 4.33484362431348564048155674336, 5.11882696749182055261433484879, 6.66717941053630213076782229667, 7.56881010058525996599806492907, 7.69562237583185992827478144333, 8.569149204901299047286666635043

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