↑

Properties

Label 2-2700-15.2-c1-0-2
Degree $2$
Conductor $2700$
Sign $-0.945 - 0.326i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3i·11-s + (−1.22 + 1.22i)13-s + 2i·19-s + (−3.67 − 3.67i)23-s − 6·29-s − 2·31-s + (1.22 + 1.22i)37-s + 6i·41-s + (2.44 − 2.44i)43-s + (−3.67 + 3.67i)47-s − 7i·49-s + (−7.34 − 7.34i)53-s − 3·59-s − 61-s + (−4.89 − 4.89i)67-s + ⋯
L(s)  = 1  + 0.904i·11-s + (−0.339 + 0.339i)13-s + 0.458i·19-s + (−0.766 − 0.766i)23-s − 1.11·29-s − 0.359·31-s + (0.201 + 0.201i)37-s + 0.937i·41-s + (0.373 − 0.373i)43-s + (−0.535 + 0.535i)47-s − i·49-s + (−1.00 − 1.00i)53-s − 0.390·59-s − 0.128·61-s + (−0.598 − 0.598i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.945 - 0.326i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4646849328\)
\(L(\frac12)\) \(\approx\) \(0.4646849328\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + (1.22 - 1.22i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (3.67 + 3.67i)T + 23iT^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-1.22 - 1.22i)T + 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-2.44 + 2.44i)T - 43iT^{2} \)
47 \( 1 + (3.67 - 3.67i)T - 47iT^{2} \)
53 \( 1 + (7.34 + 7.34i)T + 53iT^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (4.89 + 4.89i)T + 67iT^{2} \)
71 \( 1 - 15iT - 71T^{2} \)
73 \( 1 + (-4.89 + 4.89i)T - 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + (7.34 + 7.34i)T + 83iT^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (-3.67 - 3.67i)T + 97iT^{2} \)
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

−9.326011517290554470200497093944, −8.324753490773727711821844852397, −7.69207497936679716071772948314, −6.90430972349448645299730965558, −6.18852542698219678645429527370, −5.23774597809772897048031114630, −4.45291521682431894706599489522, −3.66003750556168807799370526120, −2.46132663103377576912828225484, −1.59914447358163551183372059141, 0.14303262318385794508032819983, 1.58177037647054339885980716456, 2.77316341880558834438533317099, 3.60502923864107509838873983812, 4.52262805976231171216625909526, 5.57462677875269252141415390205, 6.00021547785820543713075912417, 7.10639891425681165374925207390, 7.72414004561948279459519659138, 8.486617181621484227048017678355

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