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Properties

Label 2-330e2-1.1-c1-0-81
Degree $2$
Conductor $108900$
Sign $-1$
Analytic cond. $869.570$
Root an. cond. $29.4884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 4·13-s + 3·17-s − 5·19-s − 3·23-s − 6·29-s + 8·31-s − 7·37-s + 9·41-s − 8·43-s + 3·47-s − 6·49-s − 6·53-s − 3·59-s − 14·61-s + 2·67-s + 9·71-s − 2·73-s + 79-s + 12·83-s − 18·89-s + 4·91-s + 11·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.10·13-s + 0.727·17-s − 1.14·19-s − 0.625·23-s − 1.11·29-s + 1.43·31-s − 1.15·37-s + 1.40·41-s − 1.21·43-s + 0.437·47-s − 6/7·49-s − 0.824·53-s − 0.390·59-s − 1.79·61-s + 0.244·67-s + 1.06·71-s − 0.234·73-s + 0.112·79-s + 1.31·83-s − 1.90·89-s + 0.419·91-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(869.570\)
Root analytic conductor: \(29.4884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 108900,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good7 \( 1 - T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−13.82638950650226, −13.55721169950598, −12.90828701750993, −12.41284815278615, −12.06488538937528, −11.34534787804244, −11.03867340436949, −10.54354641748639, −10.04903359136434, −9.473639756119325, −8.947818525609557, −8.365800633176149, −8.042851396100203, −7.558287265610500, −6.813676010628640, −6.260972225300881, −5.944748014708738, −5.289371498659896, −4.612421742257011, −4.189497228988827, −3.477787668410090, −3.076017283222542, −2.101433095671654, −1.678819349302898, −0.9236809047783714, 0, 0.9236809047783714, 1.678819349302898, 2.101433095671654, 3.076017283222542, 3.477787668410090, 4.189497228988827, 4.612421742257011, 5.289371498659896, 5.944748014708738, 6.260972225300881, 6.813676010628640, 7.558287265610500, 8.042851396100203, 8.365800633176149, 8.947818525609557, 9.473639756119325, 10.04903359136434, 10.54354641748639, 11.03867340436949, 11.34534787804244, 12.06488538937528, 12.41284815278615, 12.90828701750993, 13.55721169950598, 13.82638950650226

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