Properties

Label 4-930e2-1.1-c1e2-0-5
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s − 9-s − 2·11-s + 16-s − 6·19-s − 2·20-s − 25-s − 12·29-s − 2·31-s + 36-s + 4·41-s + 2·44-s − 2·45-s − 11·49-s − 4·55-s + 28·59-s + 28·61-s − 64-s + 18·71-s + 6·76-s − 30·79-s + 2·80-s + 81-s + 2·89-s − 12·95-s + 2·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s − 1/3·9-s − 0.603·11-s + 1/4·16-s − 1.37·19-s − 0.447·20-s − 1/5·25-s − 2.22·29-s − 0.359·31-s + 1/6·36-s + 0.624·41-s + 0.301·44-s − 0.298·45-s − 1.57·49-s − 0.539·55-s + 3.64·59-s + 3.58·61-s − 1/8·64-s + 2.13·71-s + 0.688·76-s − 3.37·79-s + 0.223·80-s + 1/9·81-s + 0.211·89-s − 1.23·95-s + 0.201·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.316010079\)
\(L(\frac12)\) \(\approx\) \(1.316010079\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−10.28333496243343160019385547601, −9.811493788425270931391302038918, −9.533836145126046391854071597588, −9.027610487362500617128024115490, −8.671093947431050969266309820508, −8.106490942136047456984712173931, −8.040791268105960274438525677327, −7.24458161526450368608660492853, −6.81544539371626149443485786711, −6.49069888343439444986924777984, −5.65985739249986372350084822645, −5.55733500803930196949177533093, −5.27927439793987147849687201147, −4.50740365625074095957041183920, −3.82969445389007630682392656451, −3.73543790169156736936307645155, −2.69758095415948915636767956929, −2.21241908536177274458602143941, −1.76117529757177652857111550953, −0.52648655962142875337134174249, 0.52648655962142875337134174249, 1.76117529757177652857111550953, 2.21241908536177274458602143941, 2.69758095415948915636767956929, 3.73543790169156736936307645155, 3.82969445389007630682392656451, 4.50740365625074095957041183920, 5.27927439793987147849687201147, 5.55733500803930196949177533093, 5.65985739249986372350084822645, 6.49069888343439444986924777984, 6.81544539371626149443485786711, 7.24458161526450368608660492853, 8.040791268105960274438525677327, 8.106490942136047456984712173931, 8.671093947431050969266309820508, 9.027610487362500617128024115490, 9.533836145126046391854071597588, 9.811493788425270931391302038918, 10.28333496243343160019385547601

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