Properties

Label 4-1400e2-1.1-c1e2-0-7
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 + 2·11-s − 12·19-s + 6·29-s + 4·41-s − 49-s − 16·59-s − 20·61-s − 14·71-s + 18·79-s − 5·81-s + 8·89-s + 4·99-s − 28·101-s + 22·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 24·171-s + ⋯
L(s)  = 1  + 2/3·9-s + 0.603·11-s − 2.75·19-s + 1.11·29-s + 0.624·41-s − 1/7·49-s − 2.08·59-s − 2.56·61-s − 1.66·71-s + 2.02·79-s − 5/9·81-s + 0.847·89-s + 0.402·99-s − 2.78·101-s + 2.10·109-s − 1.72·121-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 + 0.769·169-s − 1.83·171-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

\(L(1)\) \(\approx\) \(1.749168748\)
\(L(\frac12)\) \(\approx\) \(1.749168748\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 133 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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

−9.697088823818481711701633072257, −9.272173387758320515469722329583, −9.017677189013623707133571075219, −8.595105853823862202606009325874, −8.204675737945985483454912992224, −7.65984502706950344450940204942, −7.51063017902087504211867564221, −6.66561719375102986601028406148, −6.50816979297899993781651137106, −6.28598407300574551655310136465, −5.73249939628901902563791392083, −5.07063496362356523054172266486, −4.52035633040959335322798926595, −4.22386622710022987780700289512, −4.03881938396490623702544436908, −3.03642885803833412542292928486, −2.82801833982336992243392563522, −1.74823676947384655770563592336, −1.73907920124756321652326457018, −0.53235295144991596471459908194, 0.53235295144991596471459908194, 1.73907920124756321652326457018, 1.74823676947384655770563592336, 2.82801833982336992243392563522, 3.03642885803833412542292928486, 4.03881938396490623702544436908, 4.22386622710022987780700289512, 4.52035633040959335322798926595, 5.07063496362356523054172266486, 5.73249939628901902563791392083, 6.28598407300574551655310136465, 6.50816979297899993781651137106, 6.66561719375102986601028406148, 7.51063017902087504211867564221, 7.65984502706950344450940204942, 8.204675737945985483454912992224, 8.595105853823862202606009325874, 9.017677189013623707133571075219, 9.272173387758320515469722329583, 9.697088823818481711701633072257

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