Properties

Label 4-1400e2-1.1-c1e2-0-13
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  + 5·9-s − 2·11-s − 2·19-s + 12·29-s + 8·31-s − 10·41-s − 49-s + 8·59-s + 12·61-s + 28·71-s − 28·79-s + 16·81-s + 6·89-s − 10·99-s + 20·109-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 − 10·171-s + ⋯
L(s)  = 1  + 5/3·9-s − 0.603·11-s − 0.458·19-s + 2.22·29-s + 1.43·31-s − 1.56·41-s − 1/7·49-s + 1.04·59-s + 1.53·61-s + 3.32·71-s − 3.15·79-s + 16/9·81-s + 0.635·89-s − 1.00·99-s + 1.91·109-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 − 0.764·171-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

\(L(1)\) \(\approx\) \(2.673767385\)
\(L(\frac12)\) \(\approx\) \(2.673767385\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 165 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−9.745358688199461289818865521463, −9.721398022295779136169620725401, −8.700113316836445840078054960900, −8.646280040694369929067538312951, −8.044388465311646721934293537643, −7.981934832072708442320858249336, −7.05525234841462589816395295072, −7.04401845026294622597662082657, −6.60169324507038376327337793672, −6.20906592445487140484728905006, −5.58025139286356655472514970414, −4.98777396072363524285581432063, −4.72547677431999923314001115728, −4.34382941248300225089475117596, −3.76745244395628661583181959798, −3.27803713921887626994531108069, −2.55468970728621955049122464163, −2.15448702032286839235693112090, −1.33234153155859932189094019486, −0.73397297778883513839840362463, 0.73397297778883513839840362463, 1.33234153155859932189094019486, 2.15448702032286839235693112090, 2.55468970728621955049122464163, 3.27803713921887626994531108069, 3.76745244395628661583181959798, 4.34382941248300225089475117596, 4.72547677431999923314001115728, 4.98777396072363524285581432063, 5.58025139286356655472514970414, 6.20906592445487140484728905006, 6.60169324507038376327337793672, 7.04401845026294622597662082657, 7.05525234841462589816395295072, 7.981934832072708442320858249336, 8.044388465311646721934293537643, 8.646280040694369929067538312951, 8.700113316836445840078054960900, 9.721398022295779136169620725401, 9.745358688199461289818865521463

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