Properties

Label 4-6900e2-1.1-c1e2-0-1
Degree $4$
Conductor $47610000$
Sign $1$
Analytic cond. $3035.65$
Root an. cond. $7.42272$
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  − 9-s + 8·19-s + 6·29-s − 14·31-s − 18·41-s + 13·49-s − 6·59-s − 20·61-s + 18·71-s − 16·79-s + 81-s − 6·101-s + 8·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.83·19-s + 1.11·29-s − 2.51·31-s − 2.81·41-s + 13/7·49-s − 0.781·59-s − 2.56·61-s + 2.13·71-s − 1.80·79-s + 1/9·81-s − 0.597·101-s + 0.766·109-s − 2·121-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 − 0.611·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47610000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3035.65\)
Root analytic conductor: \(7.42272\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 47610000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.595173103\)
\(L(\frac12)\) \(\approx\) \(1.595173103\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−8.040137664935669033519495330327, −7.78292729899159681713615993103, −7.31086571659068924773226946475, −7.20947023688811931026935980631, −6.86744077953363933208801249477, −6.26725986814166630022437696395, −6.12519237821493111299049909418, −5.55711147808697817417414054391, −5.19466808682169132593165654582, −5.18847283552332095693484090054, −4.61779025839317162789063680333, −4.14218874847360148194961915881, −3.61413455421682920703527380155, −3.46039571636784232692857879691, −2.89063515307166714863420891091, −2.72775880597307823600690001192, −1.84229464301911745175299685249, −1.68030843271921561787969637683, −1.05126687710298547130224223050, −0.33292646355384807422317182810, 0.33292646355384807422317182810, 1.05126687710298547130224223050, 1.68030843271921561787969637683, 1.84229464301911745175299685249, 2.72775880597307823600690001192, 2.89063515307166714863420891091, 3.46039571636784232692857879691, 3.61413455421682920703527380155, 4.14218874847360148194961915881, 4.61779025839317162789063680333, 5.18847283552332095693484090054, 5.19466808682169132593165654582, 5.55711147808697817417414054391, 6.12519237821493111299049909418, 6.26725986814166630022437696395, 6.86744077953363933208801249477, 7.20947023688811931026935980631, 7.31086571659068924773226946475, 7.78292729899159681713615993103, 8.040137664935669033519495330327

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