Properties

Label 4-1305e2-1.1-c3e2-0-0
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $5928.61$
Root an. cond. $8.77482$
Motivic weight $3$
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  − 16·4-s + 10·5-s + 52·11-s − 4·13-s + 192·16-s − 56·17-s − 36·19-s − 160·20-s − 48·23-s + 75·25-s − 58·29-s + 148·31-s + 560·37-s − 860·41-s − 80·43-s − 832·44-s + 48·47-s − 142·49-s + 64·52-s − 788·53-s + 520·55-s − 504·59-s + 764·61-s − 2.04e3·64-s − 40·65-s + 376·67-s + 896·68-s + ⋯
L(s)  = 1  − 2·4-s + 0.894·5-s + 1.42·11-s − 0.0853·13-s + 3·16-s − 0.798·17-s − 0.434·19-s − 1.78·20-s − 0.435·23-s + 3/5·25-s − 0.371·29-s + 0.857·31-s + 2.48·37-s − 3.27·41-s − 0.283·43-s − 2.85·44-s + 0.148·47-s − 0.413·49-s + 0.170·52-s − 2.04·53-s + 1.27·55-s − 1.11·59-s + 1.60·61-s − 4·64-s − 0.0763·65-s + 0.685·67-s + 1.59·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5928.61\)
Root analytic conductor: \(8.77482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.048994913\)
\(L(\frac12)\) \(\approx\) \(1.048994913\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - p T )^{2} \)
29$C_1$ \( ( 1 + p T )^{2} \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 142 T^{2} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 52 T + 1162 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 2222 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 56 T + 8434 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 36 T + 5338 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 48 T + 20014 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 74 T + p^{3} T^{2} )^{2} \)
37$D_{4}$ \( 1 - 560 T + 171002 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 860 T + 314038 T^{2} + 860 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 80 T + 141030 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 48 T + 188638 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 788 T + 444286 T^{2} + 788 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 504 T + 454678 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 764 T + 382286 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 376 T + 396966 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 444 T + p^{3} T^{2} )^{2} \)
73$D_{4}$ \( 1 + 880 T + 603890 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 172 T + 984770 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 312 T + 928006 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1316 T + 19142 p T^{2} - 1316 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 2488 T + 3353298 T^{2} - 2488 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.366918060716617321922453589057, −9.194120446897296616014670665675, −8.614125429835568954637644817734, −8.604671815176172494168402063276, −7.86943342165550890968361293581, −7.76575654919304435285068931460, −6.74019438947617298172684004515, −6.61905935626601590061979239164, −6.03499613456474630433297616673, −5.83859663941987272668171411214, −4.97880992803406105792348489190, −4.93303465533966843251417501639, −4.23853354148076210540862811824, −4.18419036928646049597651481496, −3.35392204029433751394163061445, −3.09763011669628798153861081081, −2.10275546050933889032638045578, −1.54923141624421364404647670570, −1.03897806392003852324703658836, −0.28350005563003432406608548311, 0.28350005563003432406608548311, 1.03897806392003852324703658836, 1.54923141624421364404647670570, 2.10275546050933889032638045578, 3.09763011669628798153861081081, 3.35392204029433751394163061445, 4.18419036928646049597651481496, 4.23853354148076210540862811824, 4.93303465533966843251417501639, 4.97880992803406105792348489190, 5.83859663941987272668171411214, 6.03499613456474630433297616673, 6.61905935626601590061979239164, 6.74019438947617298172684004515, 7.76575654919304435285068931460, 7.86943342165550890968361293581, 8.604671815176172494168402063276, 8.614125429835568954637644817734, 9.194120446897296616014670665675, 9.366918060716617321922453589057

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