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Properties

Label 2-2700-15.14-c0-0-0
Degree $2$
Conductor $2700$
Sign $-0.447 - 0.894i$
Analytic cond. $1.34747$
Root an. cond. $1.16080$
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  + 2i·7-s + i·13-s − 2·19-s − 31-s − i·37-s + i·43-s − 3·49-s + 2·61-s − i·67-s + i·73-s + 79-s − 2·91-s + 2i·97-s + i·103-s + 109-s + ⋯
L(s)  = 1  + 2i·7-s + i·13-s − 2·19-s − 31-s − i·37-s + i·43-s − 3·49-s + 2·61-s − i·67-s + i·73-s + 79-s − 2·91-s + 2i·97-s + i·103-s + 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9220721639\)
\(L(\frac12)\) \(\approx\) \(0.9220721639\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.091637785423212198883693756993, −8.681202808432108185843661024514, −7.963934002418641488404220151691, −6.77359256091689445132605530201, −6.20626766168947538560534306563, −5.47505686787301888731190198415, −4.64136848189099960822936788501, −3.66752309616216248150560617331, −2.39021057010672013546513966627, −1.98235594742285009280200967455, 0.56401318666265424352534726049, 1.88673787759696423020837282265, 3.25534057957201705989848958104, 4.01643959780259350109852983031, 4.66056966403181168801545658913, 5.72113353838417469636715861883, 6.70740244205642144942380433275, 7.17440219770292500724061322933, 8.037499074387559441106608185730, 8.572446661602938616566236442309

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