Properties

Label 4-1680e2-1.1-c1e2-0-16
Degree $4$
Conductor $2822400$
Sign $1$
Analytic cond. $179.958$
Root an. cond. $3.66263$
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 − 4·7-s − 3·9-s + 12·17-s + 3·25-s + 8·35-s − 4·37-s + 12·41-s + 16·43-s + 6·45-s + 24·47-s + 9·49-s + 24·59-s + 12·63-s − 16·67-s − 16·79-s + 9·81-s − 24·85-s + 12·89-s − 12·101-s − 4·109-s − 48·119-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 9-s + 2.91·17-s + 3/5·25-s + 1.35·35-s − 0.657·37-s + 1.87·41-s + 2.43·43-s + 0.894·45-s + 3.50·47-s + 9/7·49-s + 3.12·59-s + 1.51·63-s − 1.95·67-s − 1.80·79-s + 81-s − 2.60·85-s + 1.27·89-s − 1.19·101-s − 0.383·109-s − 4.40·119-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2822400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(179.958\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2822400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.701900254\)
\(L(\frac12)\) \(\approx\) \(1.701900254\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−9.480696064539601407731812953031, −9.163571076100335377451718206971, −8.825883601165449380528531189619, −8.368801190515624475181747662042, −7.902039734334558043555125042796, −7.51555577643464559491547712804, −7.21463024851942921108605313917, −6.99531770856166885083132845989, −6.08534575734793081410296563567, −5.90498972589107391443652397480, −5.54839783082297697438556053599, −5.36558660343883412833402128804, −4.20038993135112679129094501772, −4.17110607161746032592721463974, −3.59652540692593678369843025719, −2.97299256042288683656230736871, −2.93633057642367625802991181019, −2.23283319459661454064838374185, −0.858340131635840093266484899711, −0.72206754198545490062980935577, 0.72206754198545490062980935577, 0.858340131635840093266484899711, 2.23283319459661454064838374185, 2.93633057642367625802991181019, 2.97299256042288683656230736871, 3.59652540692593678369843025719, 4.17110607161746032592721463974, 4.20038993135112679129094501772, 5.36558660343883412833402128804, 5.54839783082297697438556053599, 5.90498972589107391443652397480, 6.08534575734793081410296563567, 6.99531770856166885083132845989, 7.21463024851942921108605313917, 7.51555577643464559491547712804, 7.902039734334558043555125042796, 8.368801190515624475181747662042, 8.825883601165449380528531189619, 9.163571076100335377451718206971, 9.480696064539601407731812953031

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