Properties

Label 2-87120-1.1-c1-0-0
Degree $2$
Conductor $87120$
Sign $1$
Analytic cond. $695.656$
Root an. cond. $26.3753$
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-s − 3·7-s + 5·13-s − 7·17-s − 7·19-s + 25-s + 7·29-s − 6·31-s + 3·35-s − 5·37-s − 10·41-s + 6·43-s − 10·47-s + 2·49-s + 12·53-s − 12·61-s − 5·65-s + 2·67-s − 9·71-s + 6·73-s − 10·79-s − 13·83-s + 7·85-s − 4·89-s − 15·91-s + 7·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 1.38·13-s − 1.69·17-s − 1.60·19-s + 1/5·25-s + 1.29·29-s − 1.07·31-s + 0.507·35-s − 0.821·37-s − 1.56·41-s + 0.914·43-s − 1.45·47-s + 2/7·49-s + 1.64·53-s − 1.53·61-s − 0.620·65-s + 0.244·67-s − 1.06·71-s + 0.702·73-s − 1.12·79-s − 1.42·83-s + 0.759·85-s − 0.423·89-s − 1.57·91-s + 0.718·95-s + 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(695.656\)
Root analytic conductor: \(26.3753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{87120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08628776456\)
\(L(\frac12)\) \(\approx\) \(0.08628776456\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.71764201719905, −13.28681602870638, −13.04387060994005, −12.50773696963352, −11.97644553199319, −11.33071151901962, −10.94369460455929, −10.42138879732161, −10.11062754686588, −9.178044081865131, −8.857756311287239, −8.506559351315240, −7.994110030600725, −7.019478113402410, −6.719521275721964, −6.376656123660952, −5.811117205560563, −5.043783084758929, −4.248569292606401, −4.036947463132713, −3.314066005077852, −2.775215489378888, −2.001761771378110, −1.301446205505965, −0.09403992298394934, 0.09403992298394934, 1.301446205505965, 2.001761771378110, 2.775215489378888, 3.314066005077852, 4.036947463132713, 4.248569292606401, 5.043783084758929, 5.811117205560563, 6.376656123660952, 6.719521275721964, 7.019478113402410, 7.994110030600725, 8.506559351315240, 8.857756311287239, 9.178044081865131, 10.11062754686588, 10.42138879732161, 10.94369460455929, 11.33071151901962, 11.97644553199319, 12.50773696963352, 13.04387060994005, 13.28681602870638, 13.71764201719905

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