Properties

Label 2-1860-1860.719-c0-0-1
Degree $2$
Conductor $1860$
Sign $-0.970 + 0.242i$
Analytic cond. $0.928260$
Root an. cond. $0.963462$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 0.999·12-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (−1.5 + 0.866i)17-s − 0.999·18-s + (1.5 − 0.866i)19-s + (−0.499 − 0.866i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 0.999·12-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (−1.5 + 0.866i)17-s − 0.999·18-s + (1.5 − 0.866i)19-s + (−0.499 − 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.970 + 0.242i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ -0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.335288197\)
\(L(\frac12)\) \(\approx\) \(1.335288197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - 2T + T^{2} \)
29 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.516962193755469465597681071554, −9.001822292236751917486771600141, −8.257443535524298187723441504142, −7.31130566752892153944858990890, −6.89583390768559142421167575313, −5.77836626439740611534073563731, −4.88970235896995765524089633446, −4.15860680314738563446412743096, −3.28560333936627113487291873288, −2.62222802112097520363363005613, 0.813064567638551932593266090222, 1.88122675929907580322288410650, 3.01449912105070437135905145228, 3.78043950603885932356583474779, 4.89920555448619163445224245418, 5.49556878936933059162312860346, 6.70921969504651832706622305987, 7.42373585561881789914747067857, 8.401048761558076803549812053946, 9.214616359839847528499995282563

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