Properties

Label 2-1859-1.1-c1-0-105
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  + 2.23·2-s + 1.75·3-s + 3.00·4-s + 1.83·5-s + 3.92·6-s + 2.75·7-s + 2.25·8-s + 0.0778·9-s + 4.09·10-s − 11-s + 5.27·12-s + 6.16·14-s + 3.21·15-s − 0.973·16-s + 0.375·17-s + 0.174·18-s − 5.68·19-s + 5.50·20-s + 4.83·21-s − 2.23·22-s + 4.01·23-s + 3.95·24-s − 1.64·25-s − 5.12·27-s + 8.28·28-s + 7.76·29-s + 7.19·30-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.01·3-s + 1.50·4-s + 0.819·5-s + 1.60·6-s + 1.04·7-s + 0.796·8-s + 0.0259·9-s + 1.29·10-s − 0.301·11-s + 1.52·12-s + 1.64·14-s + 0.829·15-s − 0.243·16-s + 0.0910·17-s + 0.0410·18-s − 1.30·19-s + 1.23·20-s + 1.05·21-s − 0.477·22-s + 0.837·23-s + 0.806·24-s − 0.328·25-s − 0.986·27-s + 1.56·28-s + 1.44·29-s + 1.31·30-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

\(L(1)\) \(\approx\) \(6.859714796\)
\(L(\frac12)\) \(\approx\) \(6.859714796\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T + 2T^{2} \)
3 \( 1 - 1.75T + 3T^{2} \)
5 \( 1 - 1.83T + 5T^{2} \)
7 \( 1 - 2.75T + 7T^{2} \)
17 \( 1 - 0.375T + 17T^{2} \)
19 \( 1 + 5.68T + 19T^{2} \)
23 \( 1 - 4.01T + 23T^{2} \)
29 \( 1 - 7.76T + 29T^{2} \)
31 \( 1 - 2.18T + 31T^{2} \)
37 \( 1 + 5.55T + 37T^{2} \)
41 \( 1 - 3.12T + 41T^{2} \)
43 \( 1 - 0.219T + 43T^{2} \)
47 \( 1 - 1.49T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 7.23T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 0.473T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 0.0909T + 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + 8.72T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 8.18T + 97T^{2} \)
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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

−9.035198525228072790207395285284, −8.457488631227101227774039329281, −7.63017138391326329608067063802, −6.57229771657130956121526712840, −5.86816074140069318881962847912, −4.98975914001225764333934536959, −4.40065720335931268436773846021, −3.30966960920429062287740763080, −2.51064786568793502978675736238, −1.80087627388723909543538588668, 1.80087627388723909543538588668, 2.51064786568793502978675736238, 3.30966960920429062287740763080, 4.40065720335931268436773846021, 4.98975914001225764333934536959, 5.86816074140069318881962847912, 6.57229771657130956121526712840, 7.63017138391326329608067063802, 8.457488631227101227774039329281, 9.035198525228072790207395285284

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