Properties

Label 2-87120-1.1-c1-0-131
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·7-s + 13-s + 7·17-s + 5·19-s − 8·23-s + 25-s + 5·29-s + 10·31-s − 3·35-s + 7·37-s − 6·41-s − 10·43-s − 10·47-s + 2·49-s + 4·53-s + 4·59-s + 65-s + 6·67-s + 9·71-s − 2·73-s − 6·79-s − 7·83-s + 7·85-s − 3·91-s + 5·95-s − 14·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s + 0.277·13-s + 1.69·17-s + 1.14·19-s − 1.66·23-s + 1/5·25-s + 0.928·29-s + 1.79·31-s − 0.507·35-s + 1.15·37-s − 0.937·41-s − 1.52·43-s − 1.45·47-s + 2/7·49-s + 0.549·53-s + 0.520·59-s + 0.124·65-s + 0.733·67-s + 1.06·71-s − 0.234·73-s − 0.675·79-s − 0.768·83-s + 0.759·85-s − 0.314·91-s + 0.512·95-s − 1.42·97-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: \(1\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
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

−14.10812408967931, −13.51394124472383, −13.39184998140637, −12.61645672388042, −12.15162319070743, −11.79068886982217, −11.35976577105654, −10.32345486066923, −10.07219329634773, −9.760266449401788, −9.471031222616696, −8.414257102636930, −8.192524671208545, −7.680757837769845, −6.790466471627731, −6.515057511781798, −5.995409145420894, −5.413116576354093, −4.950606074450200, −4.083837020041149, −3.510666278329081, −3.005221503641643, −2.524989857143283, −1.480952336868155, −0.9902511968621904, 0, 0.9902511968621904, 1.480952336868155, 2.524989857143283, 3.005221503641643, 3.510666278329081, 4.083837020041149, 4.950606074450200, 5.413116576354093, 5.995409145420894, 6.515057511781798, 6.790466471627731, 7.680757837769845, 8.192524671208545, 8.414257102636930, 9.471031222616696, 9.760266449401788, 10.07219329634773, 10.32345486066923, 11.35976577105654, 11.79068886982217, 12.15162319070743, 12.61645672388042, 13.39184998140637, 13.51394124472383, 14.10812408967931

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