↑

Properties

Label 2-1859-143.120-c0-0-3
Degree $2$
Conductor $1859$
Sign $-0.562 - 0.826i$
Analytic cond. $0.927761$
Root an. cond. $0.963203$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 + 1.56i)3-s + (−0.5 + 0.866i)4-s + 1.24·5-s + (−1.12 + 1.94i)9-s + (−0.5 − 0.866i)11-s − 1.80·12-s + (1.12 + 1.94i)15-s + (−0.499 − 0.866i)16-s + (−0.623 + 1.07i)20-s + (0.222 + 0.385i)23-s + 0.554·25-s − 2.24·27-s − 0.445·31-s + (0.900 − 1.56i)33-s + (−1.12 − 1.94i)36-s + (0.900 + 1.56i)37-s + ⋯
L(s)  = 1  + (0.900 + 1.56i)3-s + (−0.5 + 0.866i)4-s + 1.24·5-s + (−1.12 + 1.94i)9-s + (−0.5 − 0.866i)11-s − 1.80·12-s + (1.12 + 1.94i)15-s + (−0.499 − 0.866i)16-s + (−0.623 + 1.07i)20-s + (0.222 + 0.385i)23-s + 0.554·25-s − 2.24·27-s − 0.445·31-s + (0.900 − 1.56i)33-s + (−1.12 − 1.94i)36-s + (0.900 + 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.562 - 0.826i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -0.562 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.586588726\)
\(L(\frac12)\) \(\approx\) \(1.586588726\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - 1.24T + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + 0.445T + T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.576150188634891730935630553747, −9.067714246672446538880540634628, −8.352233928716180979930525725141, −7.79028112029775596182876957470, −6.38682836955943377687568398664, −5.29691938381814335944742897829, −4.83208272559391528737845568005, −3.70602010995952034742027978370, −3.12710546530344206179339086507, −2.24263851967859374971117561667, 1.14993001659317504856812892757, 2.05963625945230601434157637005, 2.62958429339965577284823109184, 4.20420971249250642612988501231, 5.47373326266426480485665614388, 5.96900656836973714136478068863, 6.86651280201875420621896040758, 7.46480194999476868594992714287, 8.495735577659945995505669953133, 9.097555725450653611160948867563

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