Properties

Label 4-300e2-1.1-c1e2-0-4
Degree $4$
Conductor $90000$
Sign $1$
Analytic cond. $5.73847$
Root an. cond. $1.54774$
Motivic weight $1$
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-s + 12·11-s − 10·19-s + 12·29-s − 2·31-s + 13·49-s + 12·59-s − 26·61-s − 16·79-s + 81-s − 12·99-s − 24·101-s + 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 10·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/3·9-s + 3.61·11-s − 2.29·19-s + 2.22·29-s − 0.359·31-s + 13/7·49-s + 1.56·59-s − 3.32·61-s − 1.80·79-s + 1/9·81-s − 1.20·99-s − 2.38·101-s + 1.34·109-s + 7.81·121-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 + 1/13·169-s + 0.764·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(90000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5.73847\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 90000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.761910488\)
\(L(\frac12)\) \(\approx\) \(1.761910488\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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

−11.89775426919520103720214312700, −11.71306478234434015482116738004, −10.97390440400319704438721262985, −10.75249607353648839694489368388, −10.02826616092039816278803189986, −9.623017114521372743848486422670, −8.908658711272978093623584247637, −8.724756441585117969723478982766, −8.557247793109954747773763479036, −7.60226271306038584194672676009, −6.76323999610163864840394137867, −6.71286173103656458170818552512, −6.18398257820960945511517870165, −5.74533468177333226841672193430, −4.50818170756140184808639903899, −4.31922017549397217704922497208, −3.82815000832242875890834551587, −2.96711029637426858672282506267, −1.94481564146603068110691641447, −1.12681038010934193104071281744, 1.12681038010934193104071281744, 1.94481564146603068110691641447, 2.96711029637426858672282506267, 3.82815000832242875890834551587, 4.31922017549397217704922497208, 4.50818170756140184808639903899, 5.74533468177333226841672193430, 6.18398257820960945511517870165, 6.71286173103656458170818552512, 6.76323999610163864840394137867, 7.60226271306038584194672676009, 8.557247793109954747773763479036, 8.724756441585117969723478982766, 8.908658711272978093623584247637, 9.623017114521372743848486422670, 10.02826616092039816278803189986, 10.75249607353648839694489368388, 10.97390440400319704438721262985, 11.71306478234434015482116738004, 11.89775426919520103720214312700

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