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Properties

Label 4-1400e2-1.1-c1e2-0-24
Degree $4$
Conductor $1960000$
Sign $-1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·7-s + 8-s − 9-s − 2·11-s − 4·14-s + 16-s − 18-s − 2·22-s + 4·23-s − 4·28-s − 4·29-s + 32-s − 36-s − 2·44-s + 4·46-s + 9·49-s − 4·53-s − 4·56-s − 4·58-s + 4·63-s + 64-s + 10·67-s + 12·71-s − 72-s + 8·77-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 1/3·9-s − 0.603·11-s − 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.426·22-s + 0.834·23-s − 0.755·28-s − 0.742·29-s + 0.176·32-s − 1/6·36-s − 0.301·44-s + 0.589·46-s + 9/7·49-s − 0.549·53-s − 0.534·56-s − 0.525·58-s + 0.503·63-s + 1/8·64-s + 1.22·67-s + 1.42·71-s − 0.117·72-s + 0.911·77-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 - T \)
5 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.c_h
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.17.a_v
19$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.19.a_b
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.ae_o
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.e_bu
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.41.a_ad
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.e_ec
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.61.a_ac
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.ak_dr
71$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.71.am_gg
73$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.73.a_bd
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.ai_fi
83$C_2^2$ \( 1 + 129 T^{2} + p^{2} T^{4} \) 2.83.a_ez
89$C_2^2$ \( 1 - 99 T^{2} + p^{2} T^{4} \) 2.89.a_adv
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−7.38787261454606063824909390845, −7.05661726985314940289911207707, −6.63101012373000169922908793225, −6.27853015621919985423828416919, −5.80818454195660626627345044124, −5.45709493190369287892955487961, −4.95998277297101639819209746567, −4.54815075003214761659743628409, −3.75843548036157819163396675022, −3.55416903743644122901949430045, −3.03068371059766110177738897511, −2.54598630300543587905713765785, −2.03122839640133389310484445561, −0.980929890829968063665273880393, 0, 0.980929890829968063665273880393, 2.03122839640133389310484445561, 2.54598630300543587905713765785, 3.03068371059766110177738897511, 3.55416903743644122901949430045, 3.75843548036157819163396675022, 4.54815075003214761659743628409, 4.95998277297101639819209746567, 5.45709493190369287892955487961, 5.80818454195660626627345044124, 6.27853015621919985423828416919, 6.63101012373000169922908793225, 7.05661726985314940289911207707, 7.38787261454606063824909390845

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