Properties

Degree $2$
Conductor $73920$
Sign $1$
Motivic weight $1$
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 − 11-s + 2·13-s − 15-s − 6·17-s + 4·19-s − 21-s + 8·23-s + 25-s + 27-s − 6·29-s + 8·31-s − 33-s + 35-s − 6·37-s + 2·39-s − 6·41-s + 4·43-s − 45-s − 8·47-s + 49-s − 6·51-s + 10·53-s + 55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s + 0.169·35-s − 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.37·53-s + 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73920\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{73920} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 73920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.517223913\)
\(L(\frac12)\) \(\approx\) \(2.517223913\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
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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

−13.97194641616559, −13.40776027440376, −13.24542339054833, −12.79400403958467, −12.04625928882921, −11.48231772468964, −11.20669339215430, −10.51768329972616, −10.06091296787078, −9.427812922374326, −8.941985424452384, −8.501078993851538, −8.108747047856514, −7.245834463574949, −6.977769154844331, −6.520582820530161, −5.675779260555814, −5.054758716052519, −4.611619372034779, −3.700151729501040, −3.521844351912288, −2.694295054614559, −2.213406518997110, −1.260548726735749, −0.5340232100271659, 0.5340232100271659, 1.260548726735749, 2.213406518997110, 2.694295054614559, 3.521844351912288, 3.700151729501040, 4.611619372034779, 5.054758716052519, 5.675779260555814, 6.520582820530161, 6.977769154844331, 7.245834463574949, 8.108747047856514, 8.501078993851538, 8.941985424452384, 9.427812922374326, 10.06091296787078, 10.51768329972616, 11.20669339215430, 11.48231772468964, 12.04625928882921, 12.79400403958467, 13.24542339054833, 13.40776027440376, 13.97194641616559

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