Properties

Label 4-627200-1.1-c1e2-0-7
Degree $4$
Conductor $627200$
Sign $1$
Analytic cond. $39.9908$
Root an. cond. $2.51472$
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  + 2·5-s − 6·9-s + 4·13-s − 25-s + 16·31-s − 4·37-s + 4·41-s − 8·43-s − 12·45-s + 49-s + 12·53-s + 8·65-s − 8·67-s − 16·71-s + 32·79-s + 27·81-s + 16·83-s − 12·89-s − 24·107-s − 24·117-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s + 1.10·13-s − 1/5·25-s + 2.87·31-s − 0.657·37-s + 0.624·41-s − 1.21·43-s − 1.78·45-s + 1/7·49-s + 1.64·53-s + 0.992·65-s − 0.977·67-s − 1.89·71-s + 3.60·79-s + 3·81-s + 1.75·83-s − 1.27·89-s − 2.32·107-s − 2.21·117-s − 0.545·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627200\)    =    \(2^{9} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(39.9908\)
Root analytic conductor: \(2.51472\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 627200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934895884\)
\(L(\frac12)\) \(\approx\) \(1.934895884\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−8.337962811166628714642591483591, −8.188420417684287628984093773695, −7.63952492132269464968835417403, −6.80993020459566377018827508984, −6.42262084813563967264199691637, −6.09726885496974635186701931027, −5.75153399735066949598477309550, −5.22316409435338364257345412913, −4.82505689740468593236753773357, −4.01779819314783117278154269083, −3.48608597038481628967676360249, −2.79183800612725741835263061431, −2.53130111392043318002974513818, −1.65955867447744323974785535288, −0.70758707141733256209763750081, 0.70758707141733256209763750081, 1.65955867447744323974785535288, 2.53130111392043318002974513818, 2.79183800612725741835263061431, 3.48608597038481628967676360249, 4.01779819314783117278154269083, 4.82505689740468593236753773357, 5.22316409435338364257345412913, 5.75153399735066949598477309550, 6.09726885496974635186701931027, 6.42262084813563967264199691637, 6.80993020459566377018827508984, 7.63952492132269464968835417403, 8.188420417684287628984093773695, 8.337962811166628714642591483591

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