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Properties

Label 4-50400-1.1-c1e2-0-2
Degree $4$
Conductor $50400$
Sign $1$
Analytic cond. $3.21354$
Root an. cond. $1.33889$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s + 9·11-s + 12-s + 16-s − 3·17-s − 2·18-s + 9·22-s + 24-s − 5·25-s − 5·27-s + 32-s + 9·33-s − 3·34-s − 2·36-s − 2·43-s + 9·44-s + 48-s − 6·49-s − 5·50-s − 3·51-s − 3·53-s − 5·54-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 2.71·11-s + 0.288·12-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 1.91·22-s + 0.204·24-s − 25-s − 0.962·27-s + 0.176·32-s + 1.56·33-s − 0.514·34-s − 1/3·36-s − 0.304·43-s + 1.35·44-s + 0.144·48-s − 6/7·49-s − 0.707·50-s − 0.420·51-s − 0.412·53-s − 0.680·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50400\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(3.21354\)
Root analytic conductor: \(1.33889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.649758048\)
\(L(\frac12)\) \(\approx\) \(2.649758048\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
good11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.11.aj_bo
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.19.a_h
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.23.a_t
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.31.a_ai
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.37.a_au
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.c_g
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.d_ac
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.59.am_fp
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.c_bq
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ab_ek
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \) 2.79.a_acz
83$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.83.a_n
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−9.837805864150983374415041272352, −9.595422395044950827576869638411, −9.056420047783342146862244122797, −8.576944104130893378804206694891, −8.158191467611284198740926217790, −7.34555976159783300142720265375, −6.83063986521520442850999308123, −6.30882122383555206511598620818, −5.95597642234949895812770431083, −5.12631613328249706528018359451, −4.31327260549246650079458482277, −3.84156191758560837632232001048, −3.36242486078456733648987242243, −2.35553045518756444117200517697, −1.51075417678832818771849318742, 1.51075417678832818771849318742, 2.35553045518756444117200517697, 3.36242486078456733648987242243, 3.84156191758560837632232001048, 4.31327260549246650079458482277, 5.12631613328249706528018359451, 5.95597642234949895812770431083, 6.30882122383555206511598620818, 6.83063986521520442850999308123, 7.34555976159783300142720265375, 8.158191467611284198740926217790, 8.576944104130893378804206694891, 9.056420047783342146862244122797, 9.595422395044950827576869638411, 9.837805864150983374415041272352

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