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Properties

Label 2-2700-3.2-c0-0-0
Degree $2$
Conductor $2700$
Sign $1$
Analytic cond. $1.34747$
Root an. cond. $1.16080$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 13-s + 2·19-s − 31-s + 37-s + 43-s + 3·49-s + 2·61-s + 67-s + 73-s − 79-s − 2·91-s − 2·97-s + 103-s − 109-s + ⋯
L(s)  = 1  − 2·7-s + 13-s + 2·19-s − 31-s + 37-s + 43-s + 3·49-s + 2·61-s + 67-s + 73-s − 79-s − 2·91-s − 2·97-s + 103-s − 109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2700} (701, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024987231\)
\(L(\frac12)\) \(\approx\) \(1.024987231\)
\(L(1)\) 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 \( ( 1 + T )^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 - T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( 1 - T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
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

−9.369894900461193934825877715407, −8.315847245919080541688146615858, −7.33552088663502316587376717662, −6.76350900263612993765666679587, −5.93419497994444199209392128140, −5.40533598838664707644239047412, −3.95439187298776558510360491013, −3.41610161486337847990248745465, −2.59911500477229605674785362844, −0.947520220347311601026913039911, 0.947520220347311601026913039911, 2.59911500477229605674785362844, 3.41610161486337847990248745465, 3.95439187298776558510360491013, 5.40533598838664707644239047412, 5.93419497994444199209392128140, 6.76350900263612993765666679587, 7.33552088663502316587376717662, 8.315847245919080541688146615858, 9.369894900461193934825877715407

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