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Properties

Label 2-2700-3.2-c0-0-1
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  + 7-s + 13-s − 19-s + 2·31-s + 37-s − 2·43-s − 61-s + 67-s + 73-s − 79-s + 91-s + 97-s + 103-s + 2·109-s + ⋯
L(s)  = 1  + 7-s + 13-s − 19-s + 2·31-s + 37-s − 2·43-s − 61-s + 67-s + 73-s − 79-s + 91-s + 97-s + 103-s + 2·109-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

\(L(\frac{1}{2})\) \(\approx\) \(1.423954843\)
\(L(\frac12)\) \(\approx\) \(1.423954843\)
\(L(1)\) 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 \( 1 - T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 - T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )^{2} \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
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 \( 1 - T + T^{2} \)
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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

−8.754644081605969587113585440394, −8.333350992389371529799634442428, −7.70360555381653242861203574239, −6.58233204856989175702754555454, −6.09167124349515989107413206348, −4.97310469955862315867529462787, −4.40777458705173725782031378483, −3.40686136555111625475175181240, −2.26024514438712231202783028769, −1.22564836106463277941432550181, 1.22564836106463277941432550181, 2.26024514438712231202783028769, 3.40686136555111625475175181240, 4.40777458705173725782031378483, 4.97310469955862315867529462787, 6.09167124349515989107413206348, 6.58233204856989175702754555454, 7.70360555381653242861203574239, 8.333350992389371529799634442428, 8.754644081605969587113585440394

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