Properties

Label 4-1400e2-1.1-c1e2-0-23
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  + 2·9-s + 10·11-s + 4·19-s + 6·29-s + 8·31-s − 4·41-s − 49-s + 8·59-s − 12·61-s + 10·71-s + 26·79-s − 5·81-s + 20·99-s − 36·101-s − 10·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 8·171-s + ⋯
L(s)  = 1  + 2/3·9-s + 3.01·11-s + 0.917·19-s + 1.11·29-s + 1.43·31-s − 0.624·41-s − 1/7·49-s + 1.04·59-s − 1.53·61-s + 1.18·71-s + 2.92·79-s − 5/9·81-s + 2.01·99-s − 3.58·101-s − 0.957·109-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 + 2·169-s + 0.611·171-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

\(L(1)\) \(\approx\) \(3.628979247\)
\(L(\frac12)\) \(\approx\) \(3.628979247\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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

−9.476163015046336707780206413887, −9.412948877245156641979279347824, −9.215407947505280473054234483484, −8.539009840711224179290071817799, −8.160754162375358366150802777326, −7.899487964321490089040926769421, −7.02073894909483035862929331929, −6.96301986838725771611435179009, −6.47038276679645969553838859863, −6.34985073221161005623696649485, −5.66404157382360935032578202970, −5.12586281933614860100732679464, −4.52572820501920174969384612198, −4.28063470076894864834172222196, −3.74444079350678939419962713899, −3.38103360156599868723252100317, −2.73022877991022426297831627029, −1.91263005328456586204856707282, −1.23338929273275849222745481597, −0.970384414703730431761788835925, 0.970384414703730431761788835925, 1.23338929273275849222745481597, 1.91263005328456586204856707282, 2.73022877991022426297831627029, 3.38103360156599868723252100317, 3.74444079350678939419962713899, 4.28063470076894864834172222196, 4.52572820501920174969384612198, 5.12586281933614860100732679464, 5.66404157382360935032578202970, 6.34985073221161005623696649485, 6.47038276679645969553838859863, 6.96301986838725771611435179009, 7.02073894909483035862929331929, 7.899487964321490089040926769421, 8.160754162375358366150802777326, 8.539009840711224179290071817799, 9.215407947505280473054234483484, 9.412948877245156641979279347824, 9.476163015046336707780206413887

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