Properties

Label 2-88200-1.1-c1-0-86
Degree $2$
Conductor $88200$
Sign $1$
Analytic cond. $704.280$
Root an. cond. $26.5382$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·11-s + 2·13-s + 6·17-s + 2·19-s − 5·23-s + 5·29-s + 4·31-s + 37-s − 12·41-s + 5·43-s + 2·47-s − 14·53-s + 2·59-s − 5·67-s + 9·71-s − 10·73-s + 11·79-s + 16·83-s − 14·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.50·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.04·23-s + 0.928·29-s + 0.718·31-s + 0.164·37-s − 1.87·41-s + 0.762·43-s + 0.291·47-s − 1.92·53-s + 0.260·59-s − 0.610·67-s + 1.06·71-s − 1.17·73-s + 1.23·79-s + 1.75·83-s − 1.48·89-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88200\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(704.280\)
Root analytic conductor: \(26.5382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 88200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.451075850\)
\(L(\frac12)\) \(\approx\) \(3.451075850\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.92728537838154, −13.62707059034324, −12.81946008760880, −12.21086313953925, −11.97644525929100, −11.60548413472942, −10.92516529158320, −10.35964677253864, −9.827190017462809, −9.505986576026217, −8.888750158391086, −8.263185097400190, −7.974872287032737, −7.260564400166713, −6.677535814447108, −6.170394620790393, −5.830838158732695, −5.007512184788632, −4.523097394197315, −3.735962241975457, −3.465827749450895, −2.774221747074062, −1.794548761708685, −1.299853369812115, −0.6488482981373373, 0.6488482981373373, 1.299853369812115, 1.794548761708685, 2.774221747074062, 3.465827749450895, 3.735962241975457, 4.523097394197315, 5.007512184788632, 5.830838158732695, 6.170394620790393, 6.677535814447108, 7.260564400166713, 7.974872287032737, 8.263185097400190, 8.888750158391086, 9.505986576026217, 9.827190017462809, 10.35964677253864, 10.92516529158320, 11.60548413472942, 11.97644525929100, 12.21086313953925, 12.81946008760880, 13.62707059034324, 13.92728537838154

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