Properties

Degree $2$
Conductor $106560$
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 − 4·11-s + 2·13-s − 2·17-s + 4·19-s − 8·23-s + 25-s − 2·29-s − 8·31-s − 37-s − 10·41-s + 12·43-s − 7·49-s + 6·53-s − 4·55-s − 4·59-s + 10·61-s + 2·65-s − 4·67-s + 8·71-s − 6·73-s + 8·79-s + 4·83-s − 2·85-s − 10·89-s + 4·95-s + 10·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.164·37-s − 1.56·41-s + 1.82·43-s − 49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s − 1.05·89-s + 0.410·95-s + 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(106560\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{106560} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 106560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.329442986\)
\(L(\frac12)\) \(\approx\) \(1.329442986\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
37 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
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

−13.73114218655164, −13.15425587728488, −12.88900642156228, −12.22081878718093, −11.80022984274791, −11.09738163867682, −10.81283984222851, −10.25320731627434, −9.744287233828012, −9.384161299740313, −8.627032701835496, −8.323122296779216, −7.584039319908623, −7.351919332705994, −6.576565758269031, −6.021597018708534, −5.505909510747214, −5.186586150466376, −4.438928837134082, −3.745273411115686, −3.317569956566171, −2.452774708641348, −2.054863614350420, −1.342062920215435, −0.3497823221564582, 0.3497823221564582, 1.342062920215435, 2.054863614350420, 2.452774708641348, 3.317569956566171, 3.745273411115686, 4.438928837134082, 5.186586150466376, 5.505909510747214, 6.021597018708534, 6.576565758269031, 7.351919332705994, 7.584039319908623, 8.323122296779216, 8.627032701835496, 9.384161299740313, 9.744287233828012, 10.25320731627434, 10.81283984222851, 11.09738163867682, 11.80022984274791, 12.22081878718093, 12.88900642156228, 13.15425587728488, 13.73114218655164

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