Properties

Label 2-1859-1.1-c3-0-381
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $109.684$
Root an. cond. $10.4730$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.11·2-s + 9.51·3-s + 8.93·4-s − 14.5·5-s + 39.1·6-s − 21.0·7-s + 3.83·8-s + 63.5·9-s − 59.7·10-s + 11·11-s + 84.9·12-s − 86.5·14-s − 138.·15-s − 55.6·16-s − 67.8·17-s + 261.·18-s + 88.0·19-s − 129.·20-s − 200.·21-s + 45.2·22-s − 118.·23-s + 36.5·24-s + 85.8·25-s + 347.·27-s − 187.·28-s − 78.5·29-s − 568.·30-s + ⋯
L(s)  = 1  + 1.45·2-s + 1.83·3-s + 1.11·4-s − 1.29·5-s + 2.66·6-s − 1.13·7-s + 0.169·8-s + 2.35·9-s − 1.88·10-s + 0.301·11-s + 2.04·12-s − 1.65·14-s − 2.37·15-s − 0.869·16-s − 0.967·17-s + 3.42·18-s + 1.06·19-s − 1.45·20-s − 2.08·21-s + 0.438·22-s − 1.07·23-s + 0.310·24-s + 0.686·25-s + 2.47·27-s − 1.26·28-s − 0.502·29-s − 3.45·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(109.684\)
Root analytic conductor: \(10.4730\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :3/2),\ -1)\)

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)$
bad11 \( 1 - 11T \)
13 \( 1 \)
good2 \( 1 - 4.11T + 8T^{2} \)
3 \( 1 - 9.51T + 27T^{2} \)
5 \( 1 + 14.5T + 125T^{2} \)
7 \( 1 + 21.0T + 343T^{2} \)
17 \( 1 + 67.8T + 4.91e3T^{2} \)
19 \( 1 - 88.0T + 6.85e3T^{2} \)
23 \( 1 + 118.T + 1.21e4T^{2} \)
29 \( 1 + 78.5T + 2.43e4T^{2} \)
31 \( 1 + 156.T + 2.97e4T^{2} \)
37 \( 1 + 372.T + 5.06e4T^{2} \)
41 \( 1 - 124.T + 6.89e4T^{2} \)
43 \( 1 + 230.T + 7.95e4T^{2} \)
47 \( 1 - 486.T + 1.03e5T^{2} \)
53 \( 1 + 556.T + 1.48e5T^{2} \)
59 \( 1 + 530.T + 2.05e5T^{2} \)
61 \( 1 - 479.T + 2.26e5T^{2} \)
67 \( 1 - 491.T + 3.00e5T^{2} \)
71 \( 1 + 563.T + 3.57e5T^{2} \)
73 \( 1 + 1.07e3T + 3.89e5T^{2} \)
79 \( 1 - 431.T + 4.93e5T^{2} \)
83 \( 1 - 171.T + 5.71e5T^{2} \)
89 \( 1 - 1.04e3T + 7.04e5T^{2} \)
97 \( 1 - 1.18e3T + 9.12e5T^{2} \)
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

−8.498885775528404016497461421411, −7.48125175846250147019259015023, −7.07643162421412313580457184027, −6.10515665822036560516762557581, −4.79236145701481226711565313330, −3.90095428839395208592954774501, −3.58977556539073347676772163856, −2.97851414492178523004408028299, −1.95079338696727170763900362043, 0, 1.95079338696727170763900362043, 2.97851414492178523004408028299, 3.58977556539073347676772163856, 3.90095428839395208592954774501, 4.79236145701481226711565313330, 6.10515665822036560516762557581, 7.07643162421412313580457184027, 7.48125175846250147019259015023, 8.498885775528404016497461421411

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