Properties

Degree $2$
Conductor $388080$
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 + 11-s + 2·13-s − 6·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s + 8·31-s + 6·37-s − 6·41-s + 4·43-s + 8·47-s + 10·53-s + 55-s + 12·59-s + 10·61-s + 2·65-s + 12·67-s − 8·71-s + 6·73-s − 8·79-s + 4·83-s − 6·85-s + 10·89-s − 4·95-s + 14·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1.37·53-s + 0.134·55-s + 1.56·59-s + 1.28·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s − 0.900·79-s + 0.439·83-s − 0.650·85-s + 1.05·89-s − 0.410·95-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388080\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{388080} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.131331524\)
\(L(\frac12)\) \(\approx\) \(4.131331524\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−12.59276143548590, −11.91410140192493, −11.47277098943671, −11.09607566746171, −10.76550484256798, −10.17450331960244, −9.823685332141706, −9.114025655747401, −8.874367998334039, −8.529725451682107, −8.020603822879071, −7.184421101238475, −6.962241980985851, −6.509629346993627, −5.975184252839127, −5.588082767682500, −4.873847560961311, −4.539587151437052, −3.934747912675240, −3.522414456650145, −2.708575006181706, −2.279966750521146, −1.876168814066713, −0.8819604290481211, −0.6559803976256983, 0.6559803976256983, 0.8819604290481211, 1.876168814066713, 2.279966750521146, 2.708575006181706, 3.522414456650145, 3.934747912675240, 4.539587151437052, 4.873847560961311, 5.588082767682500, 5.975184252839127, 6.509629346993627, 6.962241980985851, 7.184421101238475, 8.020603822879071, 8.529725451682107, 8.874367998334039, 9.114025655747401, 9.823685332141706, 10.17450331960244, 10.76550484256798, 11.09607566746171, 11.47277098943671, 11.91410140192493, 12.59276143548590

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