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  + 1.89·2-s − 6.15·3-s − 4.39·4-s − 1.87·5-s − 11.6·6-s + 2.77·7-s − 23.5·8-s + 10.8·9-s − 3.57·10-s + 5.01·11-s + 27.0·12-s − 31.2·13-s + 5.27·14-s + 11.5·15-s − 9.56·16-s + 42.4·17-s + 20.5·18-s + 2.31·19-s + 8.25·20-s − 17.0·21-s + 9.52·22-s − 10.8·23-s + 144.·24-s − 121.·25-s − 59.4·26-s + 99.3·27-s − 12.2·28-s + ⋯
L(s)  = 1  + 0.671·2-s − 1.18·3-s − 0.549·4-s − 0.168·5-s − 0.795·6-s + 0.149·7-s − 1.04·8-s + 0.401·9-s − 0.112·10-s + 0.137·11-s + 0.650·12-s − 0.667·13-s + 0.100·14-s + 0.199·15-s − 0.149·16-s + 0.605·17-s + 0.269·18-s + 0.0279·19-s + 0.0923·20-s − 0.177·21-s + 0.0922·22-s − 0.0983·23-s + 1.23·24-s − 0.971·25-s − 0.448·26-s + 0.708·27-s − 0.0823·28-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 \( 1 - 1.89T + 8T^{2} \)
3 \( 1 + 6.15T + 27T^{2} \)
5 \( 1 + 1.87T + 125T^{2} \)
7 \( 1 - 2.77T + 343T^{2} \)
11 \( 1 - 5.01T + 1.33e3T^{2} \)
13 \( 1 + 31.2T + 2.19e3T^{2} \)
17 \( 1 - 42.4T + 4.91e3T^{2} \)
19 \( 1 - 2.31T + 6.85e3T^{2} \)
23 \( 1 + 10.8T + 1.21e4T^{2} \)
29 \( 1 + 82.1T + 2.43e4T^{2} \)
31 \( 1 - 296.T + 2.97e4T^{2} \)
37 \( 1 - 186.T + 5.06e4T^{2} \)
41 \( 1 - 343.T + 6.89e4T^{2} \)
47 \( 1 + 200.T + 1.03e5T^{2} \)
53 \( 1 + 500.T + 1.48e5T^{2} \)
59 \( 1 - 613.T + 2.05e5T^{2} \)
61 \( 1 - 661.T + 2.26e5T^{2} \)
67 \( 1 - 198.T + 3.00e5T^{2} \)
71 \( 1 + 896.T + 3.57e5T^{2} \)
73 \( 1 - 619.T + 3.89e5T^{2} \)
79 \( 1 - 58.4T + 4.93e5T^{2} \)
83 \( 1 - 188.T + 5.71e5T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 - 1.02e3T + 9.12e5T^{2} \)
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

−8.380781338064059598850519336085, −7.68962540050044587738258851283, −6.51053086686347731556516287046, −5.94670407070794284006062628600, −5.17021027950032873447584759887, −4.59697067951195128384634397052, −3.71498314757095371893002884026, −2.59188813883471760638077905652, −0.954184414233699478199482101069, 0, 0.954184414233699478199482101069, 2.59188813883471760638077905652, 3.71498314757095371893002884026, 4.59697067951195128384634397052, 5.17021027950032873447584759887, 5.94670407070794284006062628600, 6.51053086686347731556516287046, 7.68962540050044587738258851283, 8.380781338064059598850519336085

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