Properties

Label 2-2700-1.1-c1-0-14
Degree $2$
Conductor $2700$
Sign $-1$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s + 7·13-s − 19-s − 4·31-s + 37-s − 8·43-s + 18·49-s − 13·61-s − 11·67-s − 17·73-s − 13·79-s − 35·91-s − 5·97-s + 7·103-s + 2·109-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.94·13-s − 0.229·19-s − 0.718·31-s + 0.164·37-s − 1.21·43-s + 18/7·49-s − 1.66·61-s − 1.34·67-s − 1.98·73-s − 1.46·79-s − 3.66·91-s − 0.507·97-s + 0.689·103-s + 0.191·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2700} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−8.759514901282035890818105943566, −7.64362636112665563331595399405, −6.74728657923342957215579477924, −6.19215897625319522417122883742, −5.65654369161366776038771212095, −4.27187623582428822445319431066, −3.50441900239116663157549394317, −2.91544699469635223918630301779, −1.43883911512169505886541205101, 0, 1.43883911512169505886541205101, 2.91544699469635223918630301779, 3.50441900239116663157549394317, 4.27187623582428822445319431066, 5.65654369161366776038771212095, 6.19215897625319522417122883742, 6.74728657923342957215579477924, 7.64362636112665563331595399405, 8.759514901282035890818105943566

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