Properties

Label 4-8280e2-1.1-c1e2-0-9
Degree $4$
Conductor $68558400$
Sign $1$
Analytic cond. $4371.34$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s + 2·11-s + 4·13-s − 5·17-s − 6·19-s + 2·23-s + 3·25-s − 13·29-s + 5·31-s − 2·35-s − 3·37-s − 11·41-s + 4·43-s − 16·47-s − 9·49-s + 3·53-s + 4·55-s − 9·59-s + 2·61-s + 8·65-s − 67-s − 11·71-s + 4·73-s − 2·77-s + 4·79-s + 13·83-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s + 0.603·11-s + 1.10·13-s − 1.21·17-s − 1.37·19-s + 0.417·23-s + 3/5·25-s − 2.41·29-s + 0.898·31-s − 0.338·35-s − 0.493·37-s − 1.71·41-s + 0.609·43-s − 2.33·47-s − 9/7·49-s + 0.412·53-s + 0.539·55-s − 1.17·59-s + 0.256·61-s + 0.992·65-s − 0.122·67-s − 1.30·71-s + 0.468·73-s − 0.227·77-s + 0.450·79-s + 1.42·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(68558400\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(4371.34\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 68558400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 11 T + 108 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 9 T + 134 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 130 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 11 T + 134 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 13 T + 170 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.56971372418466804539927384322, −7.21637256501280162113959343014, −6.66893558144063148510401773898, −6.63240962341754208822362135949, −6.25132032805455573608279391080, −6.15734314558101067821059669810, −5.51449684210610830824045835018, −5.31884386453976846140499991123, −4.72628109769820027393430279812, −4.58795126144026376387347860670, −3.98423927982216671634500167293, −3.71700512779088918920546717021, −3.30145101932707796468791180031, −2.95794496620772963152941574193, −2.19342976160115189162269008063, −2.10318712118214352601061277618, −1.38803074314310923480415015236, −1.35592565436354123882959875555, 0, 0, 1.35592565436354123882959875555, 1.38803074314310923480415015236, 2.10318712118214352601061277618, 2.19342976160115189162269008063, 2.95794496620772963152941574193, 3.30145101932707796468791180031, 3.71700512779088918920546717021, 3.98423927982216671634500167293, 4.58795126144026376387347860670, 4.72628109769820027393430279812, 5.31884386453976846140499991123, 5.51449684210610830824045835018, 6.15734314558101067821059669810, 6.25132032805455573608279391080, 6.63240962341754208822362135949, 6.66893558144063148510401773898, 7.21637256501280162113959343014, 7.56971372418466804539927384322

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