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Properties

Label 2-2700-5.3-c0-0-3
Degree $2$
Conductor $2700$
Sign $0.326 + 0.945i$
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  + (−1.22 − 1.22i)13-s − 2i·19-s + 31-s + (1.22 − 1.22i)37-s + (−1.22 − 1.22i)43-s + i·49-s + 2·61-s + (−1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s − i·79-s + (−1.22 − 1.22i)103-s − i·109-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)13-s − 2i·19-s + 31-s + (1.22 − 1.22i)37-s + (−1.22 − 1.22i)43-s + i·49-s + 2·61-s + (−1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s − i·79-s + (−1.22 − 1.22i)103-s − i·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.022667898\)
\(L(\frac12)\) \(\approx\) \(1.022667898\)
\(L(1)\) 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 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.874755362378329211570210923988, −8.096285965999797326640339820756, −7.31452364580608769833795660672, −6.75143235765701948630419566653, −5.64764382446814432977078452388, −5.01640899342579018412891002441, −4.21237754841951144783603945009, −2.95367453565322765369292523219, −2.39064801727847079635113236791, −0.66704196694718698684290898265, 1.51530294258981332002446848611, 2.49280897995484442344852675242, 3.60906187662975308672006808065, 4.48273208693932452318536227393, 5.19374372976942847305689014475, 6.27949487807107560485534147151, 6.76013539513920542603117633402, 7.84175262947324996623184494635, 8.214001265338353712597708547347, 9.333291250172549011556694294631

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