Properties

Label 2-88200-1.1-c1-0-76
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  + 4·11-s − 2·13-s − 2·17-s + 4·19-s + 10·29-s − 6·37-s − 6·41-s + 4·43-s + 8·47-s + 6·53-s − 4·59-s + 10·61-s − 4·67-s + 16·71-s − 14·73-s + 8·79-s + 4·83-s + 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.85·29-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 0.824·53-s − 0.520·59-s + 1.28·61-s − 0.488·67-s + 1.89·71-s − 1.63·73-s + 0.900·79-s + 0.439·83-s + 1.05·89-s + 1.01·97-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

\(L(1)\) \(\approx\) \(2.947079008\)
\(L(\frac12)\) \(\approx\) \(2.947079008\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.04634040805657, −13.48561624077014, −12.90563034723716, −12.22752422784929, −11.81473429341250, −11.76435084982599, −10.81566493354142, −10.44503272452243, −9.896693199493581, −9.357316005741389, −8.896467230192020, −8.467333829399405, −7.808467983869082, −7.176354908199973, −6.777844087687453, −6.323724668791089, −5.626557600395571, −5.022973121157508, −4.544089188243603, −3.876171522833680, −3.366884198767243, −2.635679100142959, −2.022523592086488, −1.193775977149467, −0.6068972067051589, 0.6068972067051589, 1.193775977149467, 2.022523592086488, 2.635679100142959, 3.366884198767243, 3.876171522833680, 4.544089188243603, 5.022973121157508, 5.626557600395571, 6.323724668791089, 6.777844087687453, 7.176354908199973, 7.808467983869082, 8.467333829399405, 8.896467230192020, 9.357316005741389, 9.896693199493581, 10.44503272452243, 10.81566493354142, 11.76435084982599, 11.81473429341250, 12.22752422784929, 12.90563034723716, 13.48561624077014, 14.04634040805657

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