Properties

Label 4-1400e2-1.1-c1e2-0-21
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·9-s + 2·11-s + 4·19-s − 2·29-s − 4·31-s + 24·41-s − 49-s + 20·59-s + 8·61-s − 6·71-s + 18·79-s + 27·81-s + 12·89-s + 12·99-s + 24·101-s − 10·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 2·9-s + 0.603·11-s + 0.917·19-s − 0.371·29-s − 0.718·31-s + 3.74·41-s − 1/7·49-s + 2.60·59-s + 1.02·61-s − 0.712·71-s + 2.02·79-s + 3·81-s + 1.27·89-s + 1.20·99-s + 2.38·101-s − 0.957·109-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 + 1.69·169-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

\(L(1)\) \(\approx\) \(3.417267674\)
\(L(\frac12)\) \(\approx\) \(3.417267674\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 133 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.599061180425758809174744402252, −9.358228634353551045451433032501, −9.206859942768083291343977409771, −8.678923886928553490258809363353, −7.84225347065163496641354453421, −7.81223633686724022463121525964, −7.28583188790213835075936085875, −7.09184273427207850064323285530, −6.39925925855325424010351243521, −6.30961418212361023393804338565, −5.43204864951465508273921149027, −5.31246900451201230939971897737, −4.52952872578545709568449537331, −4.27705929900894322708507356213, −3.63389025602634098694874082133, −3.59575598684042770727745005932, −2.40480825458192401725560936454, −2.19880435768382827313471797407, −1.19435034322577964247896542003, −0.929880260616596768860034320528, 0.929880260616596768860034320528, 1.19435034322577964247896542003, 2.19880435768382827313471797407, 2.40480825458192401725560936454, 3.59575598684042770727745005932, 3.63389025602634098694874082133, 4.27705929900894322708507356213, 4.52952872578545709568449537331, 5.31246900451201230939971897737, 5.43204864951465508273921149027, 6.30961418212361023393804338565, 6.39925925855325424010351243521, 7.09184273427207850064323285530, 7.28583188790213835075936085875, 7.81223633686724022463121525964, 7.84225347065163496641354453421, 8.678923886928553490258809363353, 9.206859942768083291343977409771, 9.358228634353551045451433032501, 9.599061180425758809174744402252

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