Properties

Label 2-1800-1.1-c3-0-12
Degree $2$
Conductor $1800$
Sign $1$
Analytic cond. $106.203$
Root an. cond. $10.3055$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.13·7-s − 20.7·11-s − 38.0·13-s − 4.61·17-s − 46.4·19-s − 17.2·23-s − 138.·29-s + 151.·31-s + 248.·37-s + 92.6·41-s − 443.·43-s + 262.·47-s − 341.·49-s + 278.·53-s + 662.·59-s + 365.·61-s + 786.·67-s − 401.·71-s − 290.·73-s + 23.4·77-s − 185.·79-s − 399.·83-s − 886.·89-s + 43.0·91-s + 48.9·97-s + 1.30e3·101-s + 625.·103-s + ⋯
L(s)  = 1  − 0.0611·7-s − 0.568·11-s − 0.811·13-s − 0.0658·17-s − 0.560·19-s − 0.156·23-s − 0.885·29-s + 0.876·31-s + 1.10·37-s + 0.352·41-s − 1.57·43-s + 0.813·47-s − 0.996·49-s + 0.721·53-s + 1.46·59-s + 0.767·61-s + 1.43·67-s − 0.671·71-s − 0.466·73-s + 0.0347·77-s − 0.263·79-s − 0.528·83-s − 1.05·89-s + 0.0496·91-s + 0.0512·97-s + 1.29·101-s + 0.598·103-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.501892911\)
\(L(\frac12)\) \(\approx\) \(1.501892911\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.13T + 343T^{2} \)
11 \( 1 + 20.7T + 1.33e3T^{2} \)
13 \( 1 + 38.0T + 2.19e3T^{2} \)
17 \( 1 + 4.61T + 4.91e3T^{2} \)
19 \( 1 + 46.4T + 6.85e3T^{2} \)
23 \( 1 + 17.2T + 1.21e4T^{2} \)
29 \( 1 + 138.T + 2.43e4T^{2} \)
31 \( 1 - 151.T + 2.97e4T^{2} \)
37 \( 1 - 248.T + 5.06e4T^{2} \)
41 \( 1 - 92.6T + 6.89e4T^{2} \)
43 \( 1 + 443.T + 7.95e4T^{2} \)
47 \( 1 - 262.T + 1.03e5T^{2} \)
53 \( 1 - 278.T + 1.48e5T^{2} \)
59 \( 1 - 662.T + 2.05e5T^{2} \)
61 \( 1 - 365.T + 2.26e5T^{2} \)
67 \( 1 - 786.T + 3.00e5T^{2} \)
71 \( 1 + 401.T + 3.57e5T^{2} \)
73 \( 1 + 290.T + 3.89e5T^{2} \)
79 \( 1 + 185.T + 4.93e5T^{2} \)
83 \( 1 + 399.T + 5.71e5T^{2} \)
89 \( 1 + 886.T + 7.04e5T^{2} \)
97 \( 1 - 48.9T + 9.12e5T^{2} \)
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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

−8.854354754357394859480019386351, −8.122851275043804212385817067493, −7.36842197749850055868258033630, −6.56974578146738333972741125221, −5.64850341091545453611964790559, −4.84599632318235103421770958640, −3.96725684460815477268727651795, −2.83257480782821165580291128850, −1.99853689085477842843752912380, −0.55632908388237333012209125563, 0.55632908388237333012209125563, 1.99853689085477842843752912380, 2.83257480782821165580291128850, 3.96725684460815477268727651795, 4.84599632318235103421770958640, 5.64850341091545453611964790559, 6.56974578146738333972741125221, 7.36842197749850055868258033630, 8.122851275043804212385817067493, 8.854354754357394859480019386351

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