Properties

Label 2-1859-1.1-c1-0-56
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  − 2.28·2-s + 1.22·3-s + 3.22·4-s + 2.50·5-s − 2.79·6-s + 4.50·7-s − 2.79·8-s − 1.50·9-s − 5.72·10-s + 11-s + 3.93·12-s − 10.2·14-s + 3.06·15-s − 0.0632·16-s − 3.22·17-s + 3.44·18-s − 2.34·19-s + 8.07·20-s + 5.50·21-s − 2.28·22-s + 5.79·23-s − 3.41·24-s + 1.28·25-s − 5.50·27-s + 14.5·28-s − 0.619·29-s − 7·30-s + ⋯
L(s)  = 1  − 1.61·2-s + 0.705·3-s + 1.61·4-s + 1.12·5-s − 1.13·6-s + 1.70·7-s − 0.987·8-s − 0.502·9-s − 1.81·10-s + 0.301·11-s + 1.13·12-s − 2.75·14-s + 0.790·15-s − 0.0158·16-s − 0.781·17-s + 0.811·18-s − 0.538·19-s + 1.80·20-s + 1.20·21-s − 0.487·22-s + 1.20·23-s − 0.696·24-s + 0.257·25-s − 1.05·27-s + 2.74·28-s − 0.115·29-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: \(0\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.505580144\)
\(L(\frac12)\) \(\approx\) \(1.505580144\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.28T + 2T^{2} \)
3 \( 1 - 1.22T + 3T^{2} \)
5 \( 1 - 2.50T + 5T^{2} \)
7 \( 1 - 4.50T + 7T^{2} \)
17 \( 1 + 3.22T + 17T^{2} \)
19 \( 1 + 2.34T + 19T^{2} \)
23 \( 1 - 5.79T + 23T^{2} \)
29 \( 1 + 0.619T + 29T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 - 5.06T + 37T^{2} \)
41 \( 1 + 8.36T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 + 4.06T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 0.144T + 61T^{2} \)
67 \( 1 - 8.72T + 67T^{2} \)
71 \( 1 + 1.71T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 1.87T + 79T^{2} \)
83 \( 1 + 5.38T + 83T^{2} \)
89 \( 1 + 8.50T + 89T^{2} \)
97 \( 1 - 0.112T + 97T^{2} \)
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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

−9.009569834858445862454723873272, −8.634420502573404156460062679816, −8.031413881648961474263964814963, −7.23331895248129230602853147070, −6.31453507395188991606358542423, −5.32272985522975071608126966034, −4.31797478588735510022180079166, −2.57921639314973295547991621131, −2.05737043556972920182119728839, −1.10232249550463505653129708393, 1.10232249550463505653129708393, 2.05737043556972920182119728839, 2.57921639314973295547991621131, 4.31797478588735510022180079166, 5.32272985522975071608126966034, 6.31453507395188991606358542423, 7.23331895248129230602853147070, 8.031413881648961474263964814963, 8.634420502573404156460062679816, 9.009569834858445862454723873272

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