Properties

Label 4-1400e2-1.1-c1e2-0-15
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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  + 5·9-s − 10·11-s + 12·19-s + 18·29-s + 16·41-s − 49-s − 16·59-s − 8·61-s + 16·71-s + 6·79-s + 16·81-s + 32·89-s − 50·99-s − 28·101-s + 22·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 60·171-s + ⋯
L(s)  = 1  + 5/3·9-s − 3.01·11-s + 2.75·19-s + 3.34·29-s + 2.49·41-s − 1/7·49-s − 2.08·59-s − 1.02·61-s + 1.89·71-s + 0.675·79-s + 16/9·81-s + 3.39·89-s − 5.02·99-s − 2.78·101-s + 2.10·109-s + 4.81·121-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 + 1.92·169-s + 4.58·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.570220864\)
\(L(\frac12)\) \(\approx\) \(2.570220864\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
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

−9.830355631542343965210369418935, −9.511146756694063072849780364616, −9.074472203004943738961237653960, −8.454319508032720097249076904898, −7.77483863312673220349810341921, −7.74152208327946008940158465062, −7.64451692575950282623666912201, −7.09863865729534740582752256860, −6.38508060345397383102242753684, −6.21194920886477618236243061206, −5.33803774843133144785432928121, −5.19725513004886246870487322093, −4.73191917989119424612534923405, −4.50306578537665513108174397465, −3.66694085477657493600548533871, −3.01517848820079930900166197188, −2.77379127881215527383193628069, −2.24087465779208734826785086378, −1.21516411902723416399131658557, −0.75426739068348235660808495487, 0.75426739068348235660808495487, 1.21516411902723416399131658557, 2.24087465779208734826785086378, 2.77379127881215527383193628069, 3.01517848820079930900166197188, 3.66694085477657493600548533871, 4.50306578537665513108174397465, 4.73191917989119424612534923405, 5.19725513004886246870487322093, 5.33803774843133144785432928121, 6.21194920886477618236243061206, 6.38508060345397383102242753684, 7.09863865729534740582752256860, 7.64451692575950282623666912201, 7.74152208327946008940158465062, 7.77483863312673220349810341921, 8.454319508032720097249076904898, 9.074472203004943738961237653960, 9.511146756694063072849780364616, 9.830355631542343965210369418935

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