Properties

Label 2-1800-1.1-c3-0-54
Degree $2$
Conductor $1800$
Sign $-1$
Analytic cond. $106.203$
Root an. cond. $10.3055$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s + 19·11-s + 12·13-s + 75·17-s − 91·19-s − 174·23-s + 272·29-s − 230·31-s − 182·37-s − 117·41-s + 372·43-s + 52·47-s − 307·49-s + 402·53-s − 312·59-s + 170·61-s + 763·67-s + 52·71-s − 981·73-s − 114·77-s + 1.05e3·79-s − 351·83-s − 799·89-s − 72·91-s + 962·97-s − 486·101-s − 1.18e3·103-s + ⋯
L(s)  = 1  − 0.323·7-s + 0.520·11-s + 0.256·13-s + 1.07·17-s − 1.09·19-s − 1.57·23-s + 1.74·29-s − 1.33·31-s − 0.808·37-s − 0.445·41-s + 1.31·43-s + 0.161·47-s − 0.895·49-s + 1.04·53-s − 0.688·59-s + 0.356·61-s + 1.39·67-s + 0.0869·71-s − 1.57·73-s − 0.168·77-s + 1.50·79-s − 0.464·83-s − 0.951·89-s − 0.0829·91-s + 1.00·97-s − 0.478·101-s − 1.13·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(106.203\)
Root analytic conductor: \(10.3055\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 - 19 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 - 75 T + p^{3} T^{2} \)
19 \( 1 + 91 T + p^{3} T^{2} \)
23 \( 1 + 174 T + p^{3} T^{2} \)
29 \( 1 - 272 T + p^{3} T^{2} \)
31 \( 1 + 230 T + p^{3} T^{2} \)
37 \( 1 + 182 T + p^{3} T^{2} \)
41 \( 1 + 117 T + p^{3} T^{2} \)
43 \( 1 - 372 T + p^{3} T^{2} \)
47 \( 1 - 52 T + p^{3} T^{2} \)
53 \( 1 - 402 T + p^{3} T^{2} \)
59 \( 1 + 312 T + p^{3} T^{2} \)
61 \( 1 - 170 T + p^{3} T^{2} \)
67 \( 1 - 763 T + p^{3} T^{2} \)
71 \( 1 - 52 T + p^{3} T^{2} \)
73 \( 1 + 981 T + p^{3} T^{2} \)
79 \( 1 - 1054 T + p^{3} T^{2} \)
83 \( 1 + 351 T + p^{3} T^{2} \)
89 \( 1 + 799 T + p^{3} T^{2} \)
97 \( 1 - 962 T + p^{3} T^{2} \)
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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

−8.500213508298165064258242570177, −7.82961085504382186770343857805, −6.82548898519736552722460282315, −6.16971713837544612592823203829, −5.36293415914825024920980475762, −4.21653862317739511182074736660, −3.55619470331926715340455219002, −2.39504386846651537337820772949, −1.29309732019959345422470719817, 0, 1.29309732019959345422470719817, 2.39504386846651537337820772949, 3.55619470331926715340455219002, 4.21653862317739511182074736660, 5.36293415914825024920980475762, 6.16971713837544612592823203829, 6.82548898519736552722460282315, 7.82961085504382186770343857805, 8.500213508298165064258242570177

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