Properties

Label 2-1859-1.1-c1-0-87
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $14.8441$
Root an. cond. $3.85281$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.22·2-s − 2.50·3-s − 0.507·4-s + 2.28·5-s − 3.06·6-s + 0.285·7-s − 3.06·8-s + 3.28·9-s + 2.79·10-s − 11-s + 1.27·12-s + 0.348·14-s − 5.72·15-s − 2.72·16-s + 0.507·17-s + 4.01·18-s + 3.95·19-s − 1.15·20-s − 0.714·21-s − 1.22·22-s − 0.0632·23-s + 7.67·24-s + 0.221·25-s − 0.714·27-s − 0.144·28-s − 10.7·29-s − 6.99·30-s + ⋯
L(s)  = 1  + 0.863·2-s − 1.44·3-s − 0.253·4-s + 1.02·5-s − 1.25·6-s + 0.107·7-s − 1.08·8-s + 1.09·9-s + 0.882·10-s − 0.301·11-s + 0.366·12-s + 0.0931·14-s − 1.47·15-s − 0.682·16-s + 0.122·17-s + 0.946·18-s + 0.906·19-s − 0.259·20-s − 0.155·21-s − 0.260·22-s − 0.0131·23-s + 1.56·24-s + 0.0443·25-s − 0.137·27-s − 0.0273·28-s − 1.99·29-s − 1.27·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(14.8441\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 1.22T + 2T^{2} \)
3 \( 1 + 2.50T + 3T^{2} \)
5 \( 1 - 2.28T + 5T^{2} \)
7 \( 1 - 0.285T + 7T^{2} \)
17 \( 1 - 0.507T + 17T^{2} \)
19 \( 1 - 3.95T + 19T^{2} \)
23 \( 1 + 0.0632T + 23T^{2} \)
29 \( 1 + 10.7T + 29T^{2} \)
31 \( 1 - 3.57T + 31T^{2} \)
37 \( 1 + 7.72T + 37T^{2} \)
41 \( 1 - 0.380T + 41T^{2} \)
43 \( 1 + 5.41T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 + 3.33T + 61T^{2} \)
67 \( 1 + 0.207T + 67T^{2} \)
71 \( 1 - 2.77T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 3.45T + 79T^{2} \)
83 \( 1 + 4.74T + 83T^{2} \)
89 \( 1 - 3.71T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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

−9.161293278010716322046388847952, −7.918021529241501660646281982119, −6.83051911991716502682174580394, −6.07633850420625022011302226494, −5.44285261731920590721213538170, −5.14915686582588892105789225291, −4.12000906399521823034943651220, −2.99297796205471142203649578613, −1.56425551608615543001246249097, 0, 1.56425551608615543001246249097, 2.99297796205471142203649578613, 4.12000906399521823034943651220, 5.14915686582588892105789225291, 5.44285261731920590721213538170, 6.07633850420625022011302226494, 6.83051911991716502682174580394, 7.918021529241501660646281982119, 9.161293278010716322046388847952

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