Properties

Label 2-1800-1.1-c3-0-57
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  + 5·7-s − 14·11-s + 13-s − 46·17-s + 19·19-s + 46·23-s − 14·29-s + 133·31-s + 258·37-s − 84·41-s − 167·43-s − 410·47-s − 318·49-s − 456·53-s + 194·59-s − 17·61-s + 653·67-s − 828·71-s + 570·73-s − 70·77-s − 552·79-s − 142·83-s + 1.10e3·89-s + 5·91-s + 841·97-s − 552·101-s − 308·103-s + ⋯
L(s)  = 1  + 0.269·7-s − 0.383·11-s + 0.0213·13-s − 0.656·17-s + 0.229·19-s + 0.417·23-s − 0.0896·29-s + 0.770·31-s + 1.14·37-s − 0.319·41-s − 0.592·43-s − 1.27·47-s − 0.927·49-s − 1.18·53-s + 0.428·59-s − 0.0356·61-s + 1.19·67-s − 1.38·71-s + 0.913·73-s − 0.103·77-s − 0.786·79-s − 0.187·83-s + 1.31·89-s + 0.00575·91-s + 0.880·97-s − 0.543·101-s − 0.294·103-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 \( 1 - 5 T + p^{3} T^{2} \)
11 \( 1 + 14 T + p^{3} T^{2} \)
13 \( 1 - T + p^{3} T^{2} \)
17 \( 1 + 46 T + p^{3} T^{2} \)
19 \( 1 - p T + p^{3} T^{2} \)
23 \( 1 - 2 p T + p^{3} T^{2} \)
29 \( 1 + 14 T + p^{3} T^{2} \)
31 \( 1 - 133 T + p^{3} T^{2} \)
37 \( 1 - 258 T + p^{3} T^{2} \)
41 \( 1 + 84 T + p^{3} T^{2} \)
43 \( 1 + 167 T + p^{3} T^{2} \)
47 \( 1 + 410 T + p^{3} T^{2} \)
53 \( 1 + 456 T + p^{3} T^{2} \)
59 \( 1 - 194 T + p^{3} T^{2} \)
61 \( 1 + 17 T + p^{3} T^{2} \)
67 \( 1 - 653 T + p^{3} T^{2} \)
71 \( 1 + 828 T + p^{3} T^{2} \)
73 \( 1 - 570 T + p^{3} T^{2} \)
79 \( 1 + 552 T + p^{3} T^{2} \)
83 \( 1 + 142 T + p^{3} T^{2} \)
89 \( 1 - 1104 T + p^{3} T^{2} \)
97 \( 1 - 841 T + p^{3} T^{2} \)
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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

−8.394062333003324819342546269676, −7.87622047089160165074485813137, −6.88344373330780905111651041619, −6.19406873968322098397404755407, −5.13901083687427881919391142961, −4.51277388064621196337264088417, −3.37899427814501525479819068000, −2.43147235528511051353944144449, −1.30291284351062520066448049432, 0, 1.30291284351062520066448049432, 2.43147235528511051353944144449, 3.37899427814501525479819068000, 4.51277388064621196337264088417, 5.13901083687427881919391142961, 6.19406873968322098397404755407, 6.88344373330780905111651041619, 7.87622047089160165074485813137, 8.394062333003324819342546269676

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