Properties

Label 4-960e2-1.1-c1e2-0-12
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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  − 3·5-s + 9-s − 13-s + 4·17-s + 4·25-s + 13·29-s + 13·37-s − 2·41-s − 3·45-s + 11·49-s − 6·53-s − 61-s + 3·65-s + 81-s − 12·85-s + 3·89-s − 16·97-s + 101-s − 16·109-s + 6·113-s − 117-s + 4·121-s + 3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.34·5-s + 1/3·9-s − 0.277·13-s + 0.970·17-s + 4/5·25-s + 2.41·29-s + 2.13·37-s − 0.312·41-s − 0.447·45-s + 11/7·49-s − 0.824·53-s − 0.128·61-s + 0.372·65-s + 1/9·81-s − 1.30·85-s + 0.317·89-s − 1.62·97-s + 0.0995·101-s − 1.53·109-s + 0.564·113-s − 0.0924·117-s + 4/11·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.583374277\)
\(L(\frac12)\) \(\approx\) \(1.583374277\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 137 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 69 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 60 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.026802014630520671653715408224, −7.87425940651087755371933156224, −7.34509195780880983806592442989, −6.93151058253605863191606391699, −6.48605081841693221194532843048, −5.97129068214749854859911765938, −5.44546533120119529000881903557, −4.82159520187157499614667480629, −4.41828049388328298747161929655, −4.09005754673361885725886372499, −3.45205137486449060724300338562, −2.90618149062613367102728869470, −2.45170243133299848654504061126, −1.31008367417830698796566025216, −0.66232951374400095322139015060, 0.66232951374400095322139015060, 1.31008367417830698796566025216, 2.45170243133299848654504061126, 2.90618149062613367102728869470, 3.45205137486449060724300338562, 4.09005754673361885725886372499, 4.41828049388328298747161929655, 4.82159520187157499614667480629, 5.44546533120119529000881903557, 5.97129068214749854859911765938, 6.48605081841693221194532843048, 6.93151058253605863191606391699, 7.34509195780880983806592442989, 7.87425940651087755371933156224, 8.026802014630520671653715408224

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