Properties

Label 4-405e2-1.1-c3e2-0-11
Degree $4$
Conductor $164025$
Sign $1$
Analytic cond. $571.007$
Root an. cond. $4.88833$
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  + 5·2-s + 8·4-s − 5·5-s + 30·7-s − 5·8-s − 25·10-s + 50·11-s + 20·13-s + 150·14-s − 25·16-s + 20·17-s − 88·19-s − 40·20-s + 250·22-s + 120·23-s + 100·26-s + 240·28-s − 50·29-s − 108·31-s − 40·32-s + 100·34-s − 150·35-s − 80·37-s − 440·38-s + 25·40-s + 400·41-s − 280·43-s + ⋯
L(s)  = 1  + 1.76·2-s + 4-s − 0.447·5-s + 1.61·7-s − 0.220·8-s − 0.790·10-s + 1.37·11-s + 0.426·13-s + 2.86·14-s − 0.390·16-s + 0.285·17-s − 1.06·19-s − 0.447·20-s + 2.42·22-s + 1.08·23-s + 0.754·26-s + 1.61·28-s − 0.320·29-s − 0.625·31-s − 0.220·32-s + 0.504·34-s − 0.724·35-s − 0.355·37-s − 1.87·38-s + 0.0988·40-s + 1.52·41-s − 0.993·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(571.007\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 164025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.390498516\)
\(L(\frac12)\) \(\approx\) \(8.390498516\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
good2$C_2^2$ \( 1 - 5 T + 17 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 30 T + 557 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 50 T + 1169 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 20 T - 1797 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 10 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 44 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 120 T + 2233 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 50 T - 21889 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 + 108 T - 18127 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 40 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 400 T + 91079 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 280 T - 1107 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 280 T - 25423 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 610 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 50 T - 202879 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 518 T + 41343 T^{2} - 518 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 180 T - 268363 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 700 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 410 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 516 T - 226783 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 660 T - 136187 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1500 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1630 T + 1744227 T^{2} - 1630 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.88321690357005489384247346627, −10.73674653544040004898615749892, −10.51756728298634723215758291633, −9.712475931479315333238813080643, −9.002437250714761817255421035697, −8.706262883731577315153167417337, −8.458067917910048444916136473812, −7.53169589096493035022991262688, −7.40885321315159823090041990850, −6.64714773366738634578107916293, −6.08544944814517231495242485933, −5.64138901843282852954837576542, −4.87282790617151212743784038035, −4.85905304328510838075582007155, −4.08291056000224760857308616581, −3.84825058831471065364550313732, −3.31269490309500802344880968338, −2.27179706799063025876196204964, −1.57133919639441978373780902946, −0.77874048376382023360994366938, 0.77874048376382023360994366938, 1.57133919639441978373780902946, 2.27179706799063025876196204964, 3.31269490309500802344880968338, 3.84825058831471065364550313732, 4.08291056000224760857308616581, 4.85905304328510838075582007155, 4.87282790617151212743784038035, 5.64138901843282852954837576542, 6.08544944814517231495242485933, 6.64714773366738634578107916293, 7.40885321315159823090041990850, 7.53169589096493035022991262688, 8.458067917910048444916136473812, 8.706262883731577315153167417337, 9.002437250714761817255421035697, 9.712475931479315333238813080643, 10.51756728298634723215758291633, 10.73674653544040004898615749892, 11.88321690357005489384247346627

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