Properties

Label 2-30e2-1.1-c3-0-17
Degree $2$
Conductor $900$
Sign $-1$
Analytic cond. $53.1017$
Root an. cond. $7.28709$
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  − 2·7-s − 30·11-s + 4·13-s + 90·17-s − 28·19-s + 120·23-s − 210·29-s − 4·31-s − 200·37-s − 240·41-s + 136·43-s − 120·47-s − 339·49-s − 30·53-s + 450·59-s − 166·61-s − 908·67-s + 1.02e3·71-s + 250·73-s + 60·77-s − 916·79-s − 1.14e3·83-s + 420·89-s − 8·91-s − 1.53e3·97-s − 450·101-s + 1.15e3·103-s + ⋯
L(s)  = 1  − 0.107·7-s − 0.822·11-s + 0.0853·13-s + 1.28·17-s − 0.338·19-s + 1.08·23-s − 1.34·29-s − 0.0231·31-s − 0.888·37-s − 0.914·41-s + 0.482·43-s − 0.372·47-s − 0.988·49-s − 0.0777·53-s + 0.992·59-s − 0.348·61-s − 1.65·67-s + 1.70·71-s + 0.400·73-s + 0.0888·77-s − 1.30·79-s − 1.50·83-s + 0.500·89-s − 0.00921·91-s − 1.60·97-s − 0.443·101-s + 1.10·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(53.1017\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 900,\ (\ :3/2),\ -1)\)

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 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 30 T + p^{3} T^{2} \)
13 \( 1 - 4 T + p^{3} T^{2} \)
17 \( 1 - 90 T + p^{3} T^{2} \)
19 \( 1 + 28 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 + 210 T + p^{3} T^{2} \)
31 \( 1 + 4 T + p^{3} T^{2} \)
37 \( 1 + 200 T + p^{3} T^{2} \)
41 \( 1 + 240 T + p^{3} T^{2} \)
43 \( 1 - 136 T + p^{3} T^{2} \)
47 \( 1 + 120 T + p^{3} T^{2} \)
53 \( 1 + 30 T + p^{3} T^{2} \)
59 \( 1 - 450 T + p^{3} T^{2} \)
61 \( 1 + 166 T + p^{3} T^{2} \)
67 \( 1 + 908 T + p^{3} T^{2} \)
71 \( 1 - 1020 T + p^{3} T^{2} \)
73 \( 1 - 250 T + p^{3} T^{2} \)
79 \( 1 + 916 T + p^{3} T^{2} \)
83 \( 1 + 1140 T + p^{3} T^{2} \)
89 \( 1 - 420 T + p^{3} T^{2} \)
97 \( 1 + 1538 T + p^{3} T^{2} \)
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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

−9.360566011440578903419724417452, −8.415945484121426135467273735065, −7.62852129667411935686815021293, −6.81546315356447911544890300444, −5.66509054031439778251665823144, −5.02866016367892011075458325654, −3.72840005090783613034758891235, −2.81203705638148999635379293601, −1.46634095558974472104394432693, 0, 1.46634095558974472104394432693, 2.81203705638148999635379293601, 3.72840005090783613034758891235, 5.02866016367892011075458325654, 5.66509054031439778251665823144, 6.81546315356447911544890300444, 7.62852129667411935686815021293, 8.415945484121426135467273735065, 9.360566011440578903419724417452

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