Properties

Label 4-810e2-1.1-c1e2-0-17
Degree $4$
Conductor $656100$
Sign $1$
Analytic cond. $41.8335$
Root an. cond. $2.54320$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·5-s + 8·11-s + 16-s + 6·19-s − 4·20-s + 11·25-s − 4·31-s + 14·41-s − 8·44-s + 13·49-s + 32·55-s + 2·59-s − 20·61-s − 64-s − 24·71-s − 6·76-s − 24·79-s + 4·80-s + 28·89-s + 24·95-s − 11·100-s + 28·101-s − 8·109-s + 26·121-s + 4·124-s + 24·125-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s + 2.41·11-s + 1/4·16-s + 1.37·19-s − 0.894·20-s + 11/5·25-s − 0.718·31-s + 2.18·41-s − 1.20·44-s + 13/7·49-s + 4.31·55-s + 0.260·59-s − 2.56·61-s − 1/8·64-s − 2.84·71-s − 0.688·76-s − 2.70·79-s + 0.447·80-s + 2.96·89-s + 2.46·95-s − 1.09·100-s + 2.78·101-s − 0.766·109-s + 2.36·121-s + 0.359·124-s + 2.14·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(41.8335\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 656100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.394860001\)
\(L(\frac12)\) \(\approx\) \(3.394860001\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.29809645607735272572450755661, −10.06949671266058891557957956984, −9.395112585049951091640184866380, −9.126742772977268265359764124888, −8.987744662502654117257571873340, −8.849607056089499511281939515492, −7.73719744187782745150853756845, −7.43288940021615252238555985045, −7.02657795766500975532345284901, −6.26387838019672419152785120954, −6.13739304196172495398857726597, −5.75676466570002603365012169768, −5.22933234993134907470849507438, −4.57585930542855432821538167649, −4.17816964090255960135934302845, −3.55633931222211454819618885847, −2.94171253430857875691715398824, −2.24652131125766958977381519140, −1.34505507512147112673146206943, −1.17670819434307394768684576934, 1.17670819434307394768684576934, 1.34505507512147112673146206943, 2.24652131125766958977381519140, 2.94171253430857875691715398824, 3.55633931222211454819618885847, 4.17816964090255960135934302845, 4.57585930542855432821538167649, 5.22933234993134907470849507438, 5.75676466570002603365012169768, 6.13739304196172495398857726597, 6.26387838019672419152785120954, 7.02657795766500975532345284901, 7.43288940021615252238555985045, 7.73719744187782745150853756845, 8.849607056089499511281939515492, 8.987744662502654117257571873340, 9.126742772977268265359764124888, 9.395112585049951091640184866380, 10.06949671266058891557957956984, 10.29809645607735272572450755661

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