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Properties

Label 2-1800-1.1-c1-0-14
Degree $2$
Conductor $1800$
Sign $-1$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  − 4·7-s + 4·11-s + 4·13-s − 6·17-s − 4·19-s − 4·23-s + 4·29-s + 4·37-s − 8·41-s − 12·47-s + 9·49-s − 2·53-s − 12·59-s + 2·61-s + 8·67-s − 8·71-s − 16·73-s − 16·77-s − 8·79-s − 8·83-s − 16·91-s − 8·97-s + 12·101-s − 4·103-s − 8·107-s + 18·109-s − 10·113-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.20·11-s + 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.742·29-s + 0.657·37-s − 1.24·41-s − 1.75·47-s + 9/7·49-s − 0.274·53-s − 1.56·59-s + 0.256·61-s + 0.977·67-s − 0.949·71-s − 1.87·73-s − 1.82·77-s − 0.900·79-s − 0.878·83-s − 1.67·91-s − 0.812·97-s + 1.19·101-s − 0.394·103-s − 0.773·107-s + 1.72·109-s − 0.940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -1)\)

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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.856152367312596665196297645949, −8.376027023252994051479037635261, −7.00033261127698738254386781739, −6.34361120603986280934301949235, −6.12217537238308833793189753087, −4.52307720874290127577618231214, −3.83461396837607245414233299885, −2.95795864143333934341503720948, −1.64066773371982504333519231076, 0, 1.64066773371982504333519231076, 2.95795864143333934341503720948, 3.83461396837607245414233299885, 4.52307720874290127577618231214, 6.12217537238308833793189753087, 6.34361120603986280934301949235, 7.00033261127698738254386781739, 8.376027023252994051479037635261, 8.856152367312596665196297645949

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