Properties

Label 4-1350e2-1.1-c1e2-0-2
Degree $4$
Conductor $1822500$
Sign $1$
Analytic cond. $116.204$
Root an. cond. $3.28326$
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  − 4-s + 16-s + 2·19-s − 12·29-s + 10·31-s − 24·41-s + 13·49-s − 12·59-s − 14·61-s − 64-s − 12·71-s − 2·76-s + 2·79-s − 12·89-s + 24·101-s + 26·109-s + 12·116-s − 22·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/4·16-s + 0.458·19-s − 2.22·29-s + 1.79·31-s − 3.74·41-s + 13/7·49-s − 1.56·59-s − 1.79·61-s − 1/8·64-s − 1.42·71-s − 0.229·76-s + 0.225·79-s − 1.27·89-s + 2.38·101-s + 2.49·109-s + 1.11·116-s − 2·121-s − 0.898·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1822500\)    =    \(2^{2} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(116.204\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1822500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.215184086\)
\(L(\frac12)\) \(\approx\) \(1.215184086\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \)
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

−9.836622120398146107072901775391, −9.389517595187708570774154348231, −8.944114795050053596617946401790, −8.659931308072713457773860968380, −8.325396380328993940725518319958, −7.58191430796751067552709408838, −7.57982628729479529230328223731, −7.03961768579907224725114094771, −6.44522956847503204492409833762, −6.10652183940650925047503430102, −5.64457833596253544159827872758, −5.03583953417591464402076865678, −4.91882461052074495254691884640, −4.17832460918002345553434728074, −3.86161969296874436345554460870, −3.09075971304660146306979084986, −2.98293020204661445779113799761, −1.88115637931122366117318753030, −1.55390017061958050955214855550, −0.45994623988403635555091252231, 0.45994623988403635555091252231, 1.55390017061958050955214855550, 1.88115637931122366117318753030, 2.98293020204661445779113799761, 3.09075971304660146306979084986, 3.86161969296874436345554460870, 4.17832460918002345553434728074, 4.91882461052074495254691884640, 5.03583953417591464402076865678, 5.64457833596253544159827872758, 6.10652183940650925047503430102, 6.44522956847503204492409833762, 7.03961768579907224725114094771, 7.57982628729479529230328223731, 7.58191430796751067552709408838, 8.325396380328993940725518319958, 8.659931308072713457773860968380, 8.944114795050053596617946401790, 9.389517595187708570774154348231, 9.836622120398146107072901775391

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