Properties

Label 2-330e2-1.1-c1-0-100
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·13-s + 17-s − 8·19-s − 9·23-s − 2·29-s − 9·31-s + 8·37-s − 10·41-s − 6·43-s − 7·47-s − 7·49-s + 3·53-s + 6·59-s + 3·61-s − 4·67-s + 12·71-s + 12·73-s − 11·79-s − 16·83-s − 4·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.554·13-s + 0.242·17-s − 1.83·19-s − 1.87·23-s − 0.371·29-s − 1.61·31-s + 1.31·37-s − 1.56·41-s − 0.914·43-s − 1.02·47-s − 49-s + 0.412·53-s + 0.781·59-s + 0.384·61-s − 0.488·67-s + 1.42·71-s + 1.40·73-s − 1.23·79-s − 1.75·83-s − 0.423·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

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

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: \(2\)
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.25260493567138, −13.65034948457492, −13.06129842583638, −12.71149160529074, −12.34868628462765, −11.57613547805752, −11.36934408287090, −10.73265721959862, −10.11568637118950, −9.841769584967315, −9.356232596852670, −8.470404478527781, −8.346709530199775, −7.784167271367389, −7.094509468714193, −6.630193688719895, −6.124001336589955, −5.554494540687742, −5.000134987270919, −4.349335591354605, −3.856396968012736, −3.354256237749369, −2.406122727710364, −2.030105558649540, −1.402307104211737, 0, 0, 1.402307104211737, 2.030105558649540, 2.406122727710364, 3.354256237749369, 3.856396968012736, 4.349335591354605, 5.000134987270919, 5.554494540687742, 6.124001336589955, 6.630193688719895, 7.094509468714193, 7.784167271367389, 8.346709530199775, 8.470404478527781, 9.356232596852670, 9.841769584967315, 10.11568637118950, 10.73265721959862, 11.36934408287090, 11.57613547805752, 12.34868628462765, 12.71149160529074, 13.06129842583638, 13.65034948457492, 14.25260493567138

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