Properties

Degree $2$
Conductor $141120$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·11-s − 2·13-s + 4·17-s − 8·23-s + 25-s − 2·31-s − 8·37-s + 2·41-s + 2·43-s − 10·47-s − 2·53-s + 2·55-s + 4·59-s + 10·61-s − 2·65-s − 2·67-s + 12·71-s + 10·73-s + 16·79-s + 16·83-s + 4·85-s − 14·89-s + 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s − 0.554·13-s + 0.970·17-s − 1.66·23-s + 1/5·25-s − 0.359·31-s − 1.31·37-s + 0.312·41-s + 0.304·43-s − 1.45·47-s − 0.274·53-s + 0.269·55-s + 0.520·59-s + 1.28·61-s − 0.248·65-s − 0.244·67-s + 1.42·71-s + 1.17·73-s + 1.80·79-s + 1.75·83-s + 0.433·85-s − 1.48·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{141120} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.314783131\)
\(L(\frac12)\) \(\approx\) \(2.314783131\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.61896603093179, −12.77742639766231, −12.37516720731116, −12.12911124929590, −11.51848518915566, −11.02854704506514, −10.42727152849554, −9.950018282059324, −9.601040514977596, −9.220348258665097, −8.429350621509125, −8.081403195111943, −7.595478293097670, −6.889767624395795, −6.520347208836738, −5.983776209379131, −5.290941471494242, −5.105923841353914, −4.228322685254081, −3.690672661586361, −3.289316163369456, −2.351724475287961, −1.975539764397644, −1.272885803071045, −0.4576576670705752, 0.4576576670705752, 1.272885803071045, 1.975539764397644, 2.351724475287961, 3.289316163369456, 3.690672661586361, 4.228322685254081, 5.105923841353914, 5.290941471494242, 5.983776209379131, 6.520347208836738, 6.889767624395795, 7.595478293097670, 8.081403195111943, 8.429350621509125, 9.220348258665097, 9.601040514977596, 9.950018282059324, 10.42727152849554, 11.02854704506514, 11.51848518915566, 12.12911124929590, 12.37516720731116, 12.77742639766231, 13.61896603093179

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