Properties

Label 4-6300e2-1.1-c1e2-0-16
Degree $4$
Conductor $39690000$
Sign $1$
Analytic cond. $2530.66$
Root an. cond. $7.09265$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 4·13-s + 4·19-s + 4·31-s − 2·37-s + 10·43-s + 3·49-s + 16·61-s − 14·67-s + 16·73-s + 10·79-s + 8·91-s − 20·97-s + 4·103-s − 2·109-s + 5·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.10·13-s + 0.917·19-s + 0.718·31-s − 0.328·37-s + 1.52·43-s + 3/7·49-s + 2.04·61-s − 1.71·67-s + 1.87·73-s + 1.12·79-s + 0.838·91-s − 2.03·97-s + 0.394·103-s − 0.191·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-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 − 1.07·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2530.66\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.544243890\)
\(L(\frac12)\) \(\approx\) \(4.544243890\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.113444027558810324798777966243, −7.979636124044077837789074450258, −7.51106661163631629379564321726, −7.25891588786795890978290659647, −6.67439656143676099955168987087, −6.59788041369182469264473123034, −6.06386182131145744565369306558, −5.61554044775219584328992664360, −5.44219256625498784791767463257, −5.08966483996453124771926940109, −4.42778807635828690546122299178, −4.38910803295400032057045263275, −3.66787753000870042916876877309, −3.63301223888656291973975169452, −2.86928509294642531952958975129, −2.66451721083818135937228720008, −1.95560350618348496187167926898, −1.62931467179059793975593563183, −0.943743877056097002824036133708, −0.66434007814031727085461619397, 0.66434007814031727085461619397, 0.943743877056097002824036133708, 1.62931467179059793975593563183, 1.95560350618348496187167926898, 2.66451721083818135937228720008, 2.86928509294642531952958975129, 3.63301223888656291973975169452, 3.66787753000870042916876877309, 4.38910803295400032057045263275, 4.42778807635828690546122299178, 5.08966483996453124771926940109, 5.44219256625498784791767463257, 5.61554044775219584328992664360, 6.06386182131145744565369306558, 6.59788041369182469264473123034, 6.67439656143676099955168987087, 7.25891588786795890978290659647, 7.51106661163631629379564321726, 7.979636124044077837789074450258, 8.113444027558810324798777966243

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