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  + 4.15·2-s − 4.65·3-s + 9.22·4-s − 19.9·5-s − 19.3·6-s + 27.8·7-s + 5.09·8-s − 5.29·9-s − 82.6·10-s − 0.733·11-s − 42.9·12-s + 67.6·13-s + 115.·14-s + 92.7·15-s − 52.6·16-s − 45.0·17-s − 21.9·18-s + 96.4·19-s − 183.·20-s − 129.·21-s − 3.04·22-s − 76.2·23-s − 23.7·24-s + 271.·25-s + 280.·26-s + 150.·27-s + 257.·28-s + ⋯
L(s)  = 1  + 1.46·2-s − 0.896·3-s + 1.15·4-s − 1.78·5-s − 1.31·6-s + 1.50·7-s + 0.224·8-s − 0.196·9-s − 2.61·10-s − 0.0200·11-s − 1.03·12-s + 1.44·13-s + 2.20·14-s + 1.59·15-s − 0.823·16-s − 0.642·17-s − 0.287·18-s + 1.16·19-s − 2.05·20-s − 1.34·21-s − 0.0294·22-s − 0.691·23-s − 0.201·24-s + 2.16·25-s + 2.11·26-s + 1.07·27-s + 1.73·28-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 \( 1 - 4.15T + 8T^{2} \)
3 \( 1 + 4.65T + 27T^{2} \)
5 \( 1 + 19.9T + 125T^{2} \)
7 \( 1 - 27.8T + 343T^{2} \)
11 \( 1 + 0.733T + 1.33e3T^{2} \)
13 \( 1 - 67.6T + 2.19e3T^{2} \)
17 \( 1 + 45.0T + 4.91e3T^{2} \)
19 \( 1 - 96.4T + 6.85e3T^{2} \)
23 \( 1 + 76.2T + 1.21e4T^{2} \)
29 \( 1 + 182.T + 2.43e4T^{2} \)
31 \( 1 + 37.9T + 2.97e4T^{2} \)
37 \( 1 - 95.8T + 5.06e4T^{2} \)
41 \( 1 - 33.5T + 6.89e4T^{2} \)
47 \( 1 - 533.T + 1.03e5T^{2} \)
53 \( 1 + 0.134T + 1.48e5T^{2} \)
59 \( 1 + 110.T + 2.05e5T^{2} \)
61 \( 1 - 533.T + 2.26e5T^{2} \)
67 \( 1 + 411.T + 3.00e5T^{2} \)
71 \( 1 + 216.T + 3.57e5T^{2} \)
73 \( 1 + 136.T + 3.89e5T^{2} \)
79 \( 1 - 81.5T + 4.93e5T^{2} \)
83 \( 1 + 926.T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.51e3T + 9.12e5T^{2} \)
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

−8.279135189358683358500881333229, −7.56841070895362436777013188906, −6.72567307530034879559249669816, −5.65849338539232844114186566457, −5.25184916702860170271364049737, −4.15145301272575547577809739930, −4.03068476797414804289513760707, −2.83611681717765570229810767776, −1.25303018022714946684945157011, 0, 1.25303018022714946684945157011, 2.83611681717765570229810767776, 4.03068476797414804289513760707, 4.15145301272575547577809739930, 5.25184916702860170271364049737, 5.65849338539232844114186566457, 6.72567307530034879559249669816, 7.56841070895362436777013188906, 8.279135189358683358500881333229

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