Properties

Degree $4$
Conductor $360000$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 8·11-s + 8·19-s + 4·29-s − 16·31-s − 12·41-s + 14·49-s − 24·59-s + 28·61-s + 16·71-s + 16·79-s + 81-s − 20·89-s + 8·99-s + 12·101-s + 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 2.41·11-s + 1.83·19-s + 0.742·29-s − 2.87·31-s − 1.87·41-s + 2·49-s − 3.12·59-s + 3.58·61-s + 1.89·71-s + 1.80·79-s + 1/9·81-s − 2.11·89-s + 0.804·99-s + 1.19·101-s + 3.44·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 360000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16440\)
\(L(\frac12)\) \(\approx\) \(1.16440\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−10.86818693027999930272164856398, −10.43717036399489941850520662707, −10.10520309252518321572473208349, −9.569372257466946800048741667272, −9.263308763439584118626108863212, −8.447633664682078280223357507142, −8.417456432206742402616774187403, −7.63359375554363916623389748321, −7.46441289009484144811939004072, −7.05659105140906665667498554743, −6.37253867183982780033524770776, −5.53888550831681015812030719963, −5.35248285851307466949531522148, −5.19886710852154865813871202859, −4.40236227739555376033432314237, −3.36608825253678448957738772155, −3.35159111890061679349962992986, −2.45504709485535561430260517619, −1.91768409377280683847734840880, −0.58653194341292848366378504939, 0.58653194341292848366378504939, 1.91768409377280683847734840880, 2.45504709485535561430260517619, 3.35159111890061679349962992986, 3.36608825253678448957738772155, 4.40236227739555376033432314237, 5.19886710852154865813871202859, 5.35248285851307466949531522148, 5.53888550831681015812030719963, 6.37253867183982780033524770776, 7.05659105140906665667498554743, 7.46441289009484144811939004072, 7.63359375554363916623389748321, 8.417456432206742402616774187403, 8.447633664682078280223357507142, 9.263308763439584118626108863212, 9.569372257466946800048741667272, 10.10520309252518321572473208349, 10.43717036399489941850520662707, 10.86818693027999930272164856398

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