Properties

Label 2-1800-1.1-c3-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·7-s − 16·11-s − 58·13-s + 38·17-s + 4·19-s − 80·23-s − 82·29-s − 8·31-s − 426·37-s + 246·41-s + 524·43-s − 464·47-s + 57·49-s − 702·53-s + 592·59-s + 574·61-s + 172·67-s − 768·71-s + 558·73-s + 320·77-s + 408·79-s + 164·83-s + 510·89-s + 1.16e3·91-s − 514·97-s − 666·101-s + 1.10e3·103-s + ⋯
L(s)  = 1  − 1.07·7-s − 0.438·11-s − 1.23·13-s + 0.542·17-s + 0.0482·19-s − 0.725·23-s − 0.525·29-s − 0.0463·31-s − 1.89·37-s + 0.937·41-s + 1.85·43-s − 1.44·47-s + 0.166·49-s − 1.81·53-s + 1.30·59-s + 1.20·61-s + 0.313·67-s − 1.28·71-s + 0.894·73-s + 0.473·77-s + 0.581·79-s + 0.216·83-s + 0.607·89-s + 1.33·91-s − 0.538·97-s − 0.656·101-s + 1.05·103-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: \(0\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9732070765\)
\(L(\frac12)\) \(\approx\) \(0.9732070765\)
\(L(\frac{5}{2})\) 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 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + 16 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
17 \( 1 - 38 T + p^{3} T^{2} \)
19 \( 1 - 4 T + p^{3} T^{2} \)
23 \( 1 + 80 T + p^{3} T^{2} \)
29 \( 1 + 82 T + p^{3} T^{2} \)
31 \( 1 + 8 T + p^{3} T^{2} \)
37 \( 1 + 426 T + p^{3} T^{2} \)
41 \( 1 - 6 p T + p^{3} T^{2} \)
43 \( 1 - 524 T + p^{3} T^{2} \)
47 \( 1 + 464 T + p^{3} T^{2} \)
53 \( 1 + 702 T + p^{3} T^{2} \)
59 \( 1 - 592 T + p^{3} T^{2} \)
61 \( 1 - 574 T + p^{3} T^{2} \)
67 \( 1 - 172 T + p^{3} T^{2} \)
71 \( 1 + 768 T + p^{3} T^{2} \)
73 \( 1 - 558 T + p^{3} T^{2} \)
79 \( 1 - 408 T + p^{3} T^{2} \)
83 \( 1 - 164 T + p^{3} T^{2} \)
89 \( 1 - 510 T + p^{3} T^{2} \)
97 \( 1 + 514 T + p^{3} T^{2} \)
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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

−9.074318905110119342882103176144, −8.007864983641051368253640551697, −7.35135888846694546466722713084, −6.56168187220534372967591344679, −5.69029420114272729736076781978, −4.91227355646318942674029090169, −3.79283716332709892735545543332, −2.96113747401364757488524797195, −2.00478377012594314105547272827, −0.43960637936323654873895743565, 0.43960637936323654873895743565, 2.00478377012594314105547272827, 2.96113747401364757488524797195, 3.79283716332709892735545543332, 4.91227355646318942674029090169, 5.69029420114272729736076781978, 6.56168187220534372967591344679, 7.35135888846694546466722713084, 8.007864983641051368253640551697, 9.074318905110119342882103176144

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