Properties

Label 2-30e2-1.1-c3-0-18
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  − 7·7-s + 54·11-s − 55·13-s − 18·17-s − 25·19-s + 18·23-s + 54·29-s − 271·31-s + 314·37-s + 360·41-s − 163·43-s − 522·47-s − 294·49-s + 36·53-s − 126·59-s + 47·61-s − 343·67-s + 1.08e3·71-s − 1.05e3·73-s − 378·77-s − 568·79-s − 1.42e3·83-s − 1.44e3·89-s + 385·91-s − 439·97-s − 828·101-s + 548·103-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.48·11-s − 1.17·13-s − 0.256·17-s − 0.301·19-s + 0.163·23-s + 0.345·29-s − 1.57·31-s + 1.39·37-s + 1.37·41-s − 0.578·43-s − 1.62·47-s − 6/7·49-s + 0.0933·53-s − 0.278·59-s + 0.0986·61-s − 0.625·67-s + 1.80·71-s − 1.68·73-s − 0.559·77-s − 0.808·79-s − 1.88·83-s − 1.71·89-s + 0.443·91-s − 0.459·97-s − 0.815·101-s + 0.524·103-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 \( 1 + p T + p^{3} T^{2} \)
11 \( 1 - 54 T + p^{3} T^{2} \)
13 \( 1 + 55 T + p^{3} T^{2} \)
17 \( 1 + 18 T + p^{3} T^{2} \)
19 \( 1 + 25 T + p^{3} T^{2} \)
23 \( 1 - 18 T + p^{3} T^{2} \)
29 \( 1 - 54 T + p^{3} T^{2} \)
31 \( 1 + 271 T + p^{3} T^{2} \)
37 \( 1 - 314 T + p^{3} T^{2} \)
41 \( 1 - 360 T + p^{3} T^{2} \)
43 \( 1 + 163 T + p^{3} T^{2} \)
47 \( 1 + 522 T + p^{3} T^{2} \)
53 \( 1 - 36 T + p^{3} T^{2} \)
59 \( 1 + 126 T + p^{3} T^{2} \)
61 \( 1 - 47 T + p^{3} T^{2} \)
67 \( 1 + 343 T + p^{3} T^{2} \)
71 \( 1 - 1080 T + p^{3} T^{2} \)
73 \( 1 + 1054 T + p^{3} T^{2} \)
79 \( 1 + 568 T + p^{3} T^{2} \)
83 \( 1 + 1422 T + p^{3} T^{2} \)
89 \( 1 + 1440 T + p^{3} T^{2} \)
97 \( 1 + 439 T + p^{3} T^{2} \)
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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

−9.443470050684727502275732320038, −8.584326934552546357656039390523, −7.48331874230766809047106608277, −6.74259937258134428263388020746, −5.93865693937674717490042953173, −4.74951131124620748484117996292, −3.89543702248029544028614716836, −2.74183560392729761193269454243, −1.48237133102346867222244711648, 0, 1.48237133102346867222244711648, 2.74183560392729761193269454243, 3.89543702248029544028614716836, 4.74951131124620748484117996292, 5.93865693937674717490042953173, 6.74259937258134428263388020746, 7.48331874230766809047106608277, 8.584326934552546357656039390523, 9.443470050684727502275732320038

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