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Properties

Label 2-105e2-1.1-c1-0-33
Degree $2$
Conductor $11025$
Sign $-1$
Analytic cond. $88.0350$
Root an. cond. $9.38270$
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  − 2·4-s + 7·13-s + 4·16-s − 7·19-s − 7·31-s + 37-s − 5·43-s − 14·52-s + 14·61-s − 8·64-s − 11·67-s + 7·73-s + 14·76-s − 13·79-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s + 1.94·13-s + 16-s − 1.60·19-s − 1.25·31-s + 0.164·37-s − 0.762·43-s − 1.94·52-s + 1.79·61-s − 64-s − 1.34·67-s + 0.819·73-s + 1.60·76-s − 1.46·79-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(88.0350\)
Root analytic conductor: \(9.38270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11025,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.71375988647405, −16.35846470714573, −15.59845818185305, −15.00084486652746, −14.49174960868833, −13.86758950985686, −13.25204381064757, −12.97124369243016, −12.39577256882688, −11.45644965958982, −10.96763192907953, −10.39934631052245, −9.759612467709412, −8.937942024296082, −8.599436641612125, −8.178799140872350, −7.264553790946686, −6.415844632298848, −5.903067051363179, −5.227559122189026, −4.328793953204915, −3.872653074450142, −3.224339270854155, −1.993602805120383, −1.121051020092006, 0, 1.121051020092006, 1.993602805120383, 3.224339270854155, 3.872653074450142, 4.328793953204915, 5.227559122189026, 5.903067051363179, 6.415844632298848, 7.264553790946686, 8.178799140872350, 8.599436641612125, 8.937942024296082, 9.759612467709412, 10.39934631052245, 10.96763192907953, 11.45644965958982, 12.39577256882688, 12.97124369243016, 13.25204381064757, 13.86758950985686, 14.49174960868833, 15.00084486652746, 15.59845818185305, 16.35846470714573, 16.71375988647405

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