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Properties

Label 2-141960-1.1-c1-0-18
Degree $2$
Conductor $141960$
Sign $1$
Analytic cond. $1133.55$
Root an. cond. $33.6683$
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  − 3-s − 5-s + 7-s + 9-s + 4·11-s + 15-s + 2·17-s + 4·19-s − 21-s + 25-s − 27-s − 10·29-s − 4·33-s − 35-s − 6·37-s + 6·41-s − 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s + 6·53-s − 4·55-s − 4·57-s + 4·59-s − 10·61-s + 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.696·33-s − 0.169·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141960\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1133.55\)
Root analytic conductor: \(33.6683\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 141960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.093894330\)
\(L(\frac12)\) \(\approx\) \(2.093894330\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
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

−13.55743110742687, −12.70561042462232, −12.29487925314901, −11.99827401186702, −11.51326644185795, −10.98253470780845, −10.80047156901080, −9.976776505550942, −9.499519807008376, −9.140990364276675, −8.592521215155008, −7.822326312665118, −7.596168181170642, −6.942334101969642, −6.587349416374634, −5.865774710141679, −5.354110779904387, −5.057743625761695, −4.112014811893025, −3.892062990283297, −3.356018779969415, −2.485196646635611, −1.710596669639812, −1.174520138799929, −0.4950546356862688, 0.4950546356862688, 1.174520138799929, 1.710596669639812, 2.485196646635611, 3.356018779969415, 3.892062990283297, 4.112014811893025, 5.057743625761695, 5.354110779904387, 5.865774710141679, 6.587349416374634, 6.942334101969642, 7.596168181170642, 7.822326312665118, 8.592521215155008, 9.140990364276675, 9.499519807008376, 9.976776505550942, 10.80047156901080, 10.98253470780845, 11.51326644185795, 11.99827401186702, 12.29487925314901, 12.70561042462232, 13.55743110742687

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