Properties

Label 4-300e2-1.1-c2e2-0-6
Degree $4$
Conductor $90000$
Sign $1$
Analytic cond. $66.8209$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 9·9-s − 22·19-s + 118·31-s − 71·49-s − 242·61-s + 284·79-s + 81·81-s − 142·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + 198·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 9-s − 1.15·19-s + 3.80·31-s − 1.44·49-s − 3.96·61-s + 3.59·79-s + 81-s − 1.30·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.13·169-s + 1.15·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(66.8209\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 90000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.525019825\)
\(L(\frac12)\) \(\approx\) \(1.525019825\)
\(L(2)\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 71 T^{2} + p^{4} T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2^2$ \( 1 + 191 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 11 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_2$ \( ( 1 - 59 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2062 T^{2} + p^{4} T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2^2$ \( 1 + 3191 T^{2} + p^{4} T^{4} \)
47$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 + 121 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 8809 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 - 8542 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2^2$ \( 1 + 9071 T^{2} + p^{4} T^{4} \)
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

−11.91893341119461219277812358815, −11.14313299260001934214916051504, −10.90882906484152064632781482593, −10.43376664125113216007951691107, −9.854263914352636454474409972376, −9.454965557907483102144459302987, −8.815381953638103562640972287861, −8.437165553783129398806432879654, −7.953258683561038491650617967762, −7.64802806328796646043794127308, −6.56719051155626428363894701120, −6.37137258114001689092462252030, −6.04468014680664510962976631743, −5.05874655275264717236550106787, −4.71065036191316452858260308615, −4.09292806883798939428159789023, −3.09567823811564449111867640423, −2.76878541219959199789426933710, −1.80714191230484323480230343056, −0.59979971103601221384346745151, 0.59979971103601221384346745151, 1.80714191230484323480230343056, 2.76878541219959199789426933710, 3.09567823811564449111867640423, 4.09292806883798939428159789023, 4.71065036191316452858260308615, 5.05874655275264717236550106787, 6.04468014680664510962976631743, 6.37137258114001689092462252030, 6.56719051155626428363894701120, 7.64802806328796646043794127308, 7.953258683561038491650617967762, 8.437165553783129398806432879654, 8.815381953638103562640972287861, 9.454965557907483102144459302987, 9.854263914352636454474409972376, 10.43376664125113216007951691107, 10.90882906484152064632781482593, 11.14313299260001934214916051504, 11.91893341119461219277812358815

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