Properties

Label 4-3800e2-1.1-c1e2-0-3
Degree $4$
Conductor $14440000$
Sign $1$
Analytic cond. $920.706$
Root an. cond. $5.50846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 4·9-s + 4·11-s + 4·13-s − 2·19-s − 4·23-s − 4·29-s + 8·31-s − 4·37-s − 4·41-s + 12·43-s + 4·47-s + 6·49-s + 4·53-s − 8·59-s − 16·63-s + 8·73-s + 16·77-s + 7·81-s − 20·83-s − 4·89-s + 16·91-s + 12·97-s − 16·99-s − 16·101-s + 24·107-s − 4·109-s + ⋯
L(s)  = 1  + 1.51·7-s − 4/3·9-s + 1.20·11-s + 1.10·13-s − 0.458·19-s − 0.834·23-s − 0.742·29-s + 1.43·31-s − 0.657·37-s − 0.624·41-s + 1.82·43-s + 0.583·47-s + 6/7·49-s + 0.549·53-s − 1.04·59-s − 2.01·63-s + 0.936·73-s + 1.82·77-s + 7/9·81-s − 2.19·83-s − 0.423·89-s + 1.67·91-s + 1.21·97-s − 1.60·99-s − 1.59·101-s + 2.32·107-s − 0.383·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14440000\)    =    \(2^{6} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(920.706\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.351275313\)
\(L(\frac12)\) \(\approx\) \(3.351275313\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 180 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.599695084661606637731054770031, −8.464301460876255247221638245994, −7.956233061665725943079831945307, −7.74656679353531021670455371625, −7.29563070466573465519863207826, −6.78000551289898388971381236249, −6.30727749883512906252452232986, −6.17916851841329268861775406504, −5.57396693586309496626958424868, −5.52571038220754506875514793610, −4.90927037695393586447708210449, −4.39882658436348028913229099451, −4.06700119125360014229047939819, −3.87608141539292505504820917889, −3.05878823464104863268574838508, −2.89620661412961236256898253385, −1.94303006789807970870348128221, −1.90708294084359614968259958260, −1.16659750013361091362565965697, −0.58048344731254869550521568010, 0.58048344731254869550521568010, 1.16659750013361091362565965697, 1.90708294084359614968259958260, 1.94303006789807970870348128221, 2.89620661412961236256898253385, 3.05878823464104863268574838508, 3.87608141539292505504820917889, 4.06700119125360014229047939819, 4.39882658436348028913229099451, 4.90927037695393586447708210449, 5.52571038220754506875514793610, 5.57396693586309496626958424868, 6.17916851841329268861775406504, 6.30727749883512906252452232986, 6.78000551289898388971381236249, 7.29563070466573465519863207826, 7.74656679353531021670455371625, 7.956233061665725943079831945307, 8.464301460876255247221638245994, 8.599695084661606637731054770031

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