Properties

Label 4-86400-1.1-c1e2-0-19
Degree $4$
Conductor $86400$
Sign $1$
Analytic cond. $5.50893$
Root an. cond. $1.53202$
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  + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s + 12-s + 2·15-s + 16-s + 18-s − 8·19-s + 2·20-s + 24-s + 3·25-s + 27-s + 12·29-s + 2·30-s + 32-s + 36-s − 8·38-s + 2·40-s − 8·43-s + 2·45-s + 48-s + 2·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.204·24-s + 3/5·25-s + 0.192·27-s + 2.22·29-s + 0.365·30-s + 0.176·32-s + 1/6·36-s − 1.29·38-s + 0.316·40-s − 1.21·43-s + 0.298·45-s + 0.144·48-s + 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86400\)    =    \(2^{7} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(5.50893\)
Root analytic conductor: \(1.53202\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.286902394\)
\(L(\frac12)\) \(\approx\) \(3.286902394\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$ \( 1 - T \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.880541566397139490214892173475, −9.082506062594783472831133921500, −8.644229019078172602684577487278, −8.307525687782695431948900879973, −7.71052407307967077644420749342, −6.76895238282715006434981339956, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −5.48025531888955534300415720953, −4.84143328212506167706839183620, −4.32687879913691587957737883725, −3.69936628544405197022333195268, −2.74628747875934588873509462774, −2.39041205999551187286748903960, −1.43294542786364492056174526795, 1.43294542786364492056174526795, 2.39041205999551187286748903960, 2.74628747875934588873509462774, 3.69936628544405197022333195268, 4.32687879913691587957737883725, 4.84143328212506167706839183620, 5.48025531888955534300415720953, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 6.76895238282715006434981339956, 7.71052407307967077644420749342, 8.307525687782695431948900879973, 8.644229019078172602684577487278, 9.082506062594783472831133921500, 9.880541566397139490214892173475

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