Properties

Label 4-810e2-1.1-c3e2-0-5
Degree $4$
Conductor $656100$
Sign $1$
Analytic cond. $2284.03$
Root an. cond. $6.91314$
Motivic weight $3$
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  − 2·2-s + 5·5-s + 22·7-s + 8·8-s − 10·10-s + 12·11-s − 38·13-s − 44·14-s − 16·16-s − 210·17-s − 314·19-s − 24·22-s + 117·23-s + 76·26-s − 66·29-s + 25·31-s + 420·34-s + 110·35-s + 628·37-s + 628·38-s + 40·40-s + 504·41-s − 380·43-s − 234·46-s + 252·47-s + 343·49-s + 6·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s + 1.18·7-s + 0.353·8-s − 0.316·10-s + 0.328·11-s − 0.810·13-s − 0.839·14-s − 1/4·16-s − 2.99·17-s − 3.79·19-s − 0.232·22-s + 1.06·23-s + 0.573·26-s − 0.422·29-s + 0.144·31-s + 2.11·34-s + 0.531·35-s + 2.79·37-s + 2.68·38-s + 0.158·40-s + 1.91·41-s − 1.34·43-s − 0.750·46-s + 0.782·47-s + 49-s + 0.0155·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2284.03\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 656100,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6174114846\)
\(L(\frac12)\) \(\approx\) \(0.6174114846\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - p T + p^{2} T^{2} \)
good7$C_2^2$ \( 1 - 22 T + 141 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 38 T - 753 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 105 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 157 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 117 T + 1522 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 66 T - 20033 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 25 T - 29166 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 314 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 504 T + 185095 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 380 T + 64893 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 252 T - 40319 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 318 T - 104255 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 293 T - 141132 T^{2} + 293 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 322 T - 197079 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 120 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 44 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 917 T + 347850 T^{2} + 917 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 309 T - 476306 T^{2} + 309 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1272 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 1328 T + 850911 T^{2} + 1328 p^{3} T^{3} + p^{6} T^{4} \)
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

−10.21329434718839759677724107285, −9.266054456721301663323836761968, −9.263873267013257558999924228014, −8.897273129830875603043271272542, −8.361200626164577819303292543625, −8.203327747336595192337309863756, −7.61571460739274506560288507202, −6.93649341578548794791700917817, −6.77685070303960724537418823328, −6.06358360132559619812362871724, −6.02708571598588955400340730069, −4.81132760650503470713071391177, −4.74402093722274902582782310628, −4.20427696449871493095796115739, −4.06516137279457721891642708526, −2.52231200453497136235064459570, −2.32503152974620651497691254362, −2.04092759959770561288955899543, −1.14008835368096043770638170340, −0.24670990485603835418528857033, 0.24670990485603835418528857033, 1.14008835368096043770638170340, 2.04092759959770561288955899543, 2.32503152974620651497691254362, 2.52231200453497136235064459570, 4.06516137279457721891642708526, 4.20427696449871493095796115739, 4.74402093722274902582782310628, 4.81132760650503470713071391177, 6.02708571598588955400340730069, 6.06358360132559619812362871724, 6.77685070303960724537418823328, 6.93649341578548794791700917817, 7.61571460739274506560288507202, 8.203327747336595192337309863756, 8.361200626164577819303292543625, 8.897273129830875603043271272542, 9.263873267013257558999924228014, 9.266054456721301663323836761968, 10.21329434718839759677724107285

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