Properties

Label 2-1859-1.1-c1-0-100
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.76·2-s − 0.0832·3-s + 5.66·4-s + 1.06·5-s − 0.230·6-s + 3.59·7-s + 10.1·8-s − 2.99·9-s + 2.95·10-s − 11-s − 0.471·12-s + 9.96·14-s − 0.0888·15-s + 16.7·16-s − 7.39·17-s − 8.28·18-s + 4.02·19-s + 6.05·20-s − 0.299·21-s − 2.76·22-s + 0.0759·23-s − 0.845·24-s − 3.86·25-s + 0.498·27-s + 20.3·28-s − 4.74·29-s − 0.246·30-s + ⋯
L(s)  = 1  + 1.95·2-s − 0.0480·3-s + 2.83·4-s + 0.477·5-s − 0.0940·6-s + 1.35·7-s + 3.59·8-s − 0.997·9-s + 0.934·10-s − 0.301·11-s − 0.136·12-s + 2.66·14-s − 0.0229·15-s + 4.19·16-s − 1.79·17-s − 1.95·18-s + 0.924·19-s + 1.35·20-s − 0.0653·21-s − 0.590·22-s + 0.0158·23-s − 0.172·24-s − 0.772·25-s + 0.0959·27-s + 3.85·28-s − 0.881·29-s − 0.0449·30-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: \(0\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.899798681\)
\(L(\frac12)\) \(\approx\) \(6.899798681\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.76T + 2T^{2} \)
3 \( 1 + 0.0832T + 3T^{2} \)
5 \( 1 - 1.06T + 5T^{2} \)
7 \( 1 - 3.59T + 7T^{2} \)
17 \( 1 + 7.39T + 17T^{2} \)
19 \( 1 - 4.02T + 19T^{2} \)
23 \( 1 - 0.0759T + 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 - 0.177T + 31T^{2} \)
37 \( 1 + 4.15T + 37T^{2} \)
41 \( 1 + 3.24T + 41T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 - 7.85T + 47T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 + 6.77T + 59T^{2} \)
61 \( 1 - 2.93T + 61T^{2} \)
67 \( 1 + 6.14T + 67T^{2} \)
71 \( 1 - 7.85T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 - 9.04T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.196650015774237891347397422985, −8.138325673633044533349368424021, −7.43580329752735500965362587704, −6.50786022447050315252623744342, −5.71591601946005567534840449419, −5.14585253908683128141996122762, −4.50702193960354438097367305323, −3.49074878307235547334560784422, −2.41190741687326156996694925023, −1.77979682211335949623030085176, 1.77979682211335949623030085176, 2.41190741687326156996694925023, 3.49074878307235547334560784422, 4.50702193960354438097367305323, 5.14585253908683128141996122762, 5.71591601946005567534840449419, 6.50786022447050315252623744342, 7.43580329752735500965362587704, 8.138325673633044533349368424021, 9.196650015774237891347397422985

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