Properties

Label 4-405e2-1.1-c2e2-0-0
Degree $4$
Conductor $164025$
Sign $1$
Analytic cond. $121.781$
Root an. cond. $3.32196$
Motivic weight $2$
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  − 2-s + 4·4-s − 5·5-s − 11·8-s + 5·10-s + 11·16-s − 28·17-s − 44·19-s − 20·20-s − 34·23-s − 2·31-s − 44·32-s + 28·34-s + 44·38-s + 55·40-s + 34·46-s + 14·47-s − 49·49-s − 172·53-s + 118·61-s + 2·62-s + 57·64-s − 112·68-s − 176·76-s − 98·79-s − 55·80-s − 154·83-s + ⋯
L(s)  = 1  − 1/2·2-s + 4-s − 5-s − 1.37·8-s + 1/2·10-s + 0.687·16-s − 1.64·17-s − 2.31·19-s − 20-s − 1.47·23-s − 0.0645·31-s − 1.37·32-s + 0.823·34-s + 1.15·38-s + 11/8·40-s + 0.739·46-s + 0.297·47-s − 49-s − 3.24·53-s + 1.93·61-s + 1/31·62-s + 0.890·64-s − 1.64·68-s − 2.31·76-s − 1.24·79-s − 0.687·80-s − 1.85·83-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(121.781\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{405} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 164025,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.03286171491\)
\(L(\frac12)\) \(\approx\) \(0.03286171491\)
\(L(2)\) 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$C_2^2$ \( 1 + T - 3 T^{2} + p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
11$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
17$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 34 T + 627 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
31$C_2^2$ \( 1 + 2 T - 957 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
43$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
47$C_2^2$ \( 1 - 14 T - 2013 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 86 T + p^{2} T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
61$C_2^2$ \( 1 - 118 T + 10203 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
79$C_2^2$ \( 1 + 98 T + 3363 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 + 154 T + 16827 T^{2} + 154 p^{2} T^{3} + p^{4} T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
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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

−11.24695729373639647678619778021, −10.87219723725438341815567761614, −10.57600315855219390884594685293, −9.792468513161543745655948759513, −9.551628077936617551967454490951, −8.764627998005545138459719887879, −8.476978157658496165256371944774, −8.223913489044351500887777074324, −7.59633877458315899320980396190, −6.99323642123612158462826833226, −6.61915776285840484272313761167, −6.16462809648141667525249024065, −5.85183929586760904875550215167, −4.72510502641948934612071710553, −4.32778382847608198057906378047, −3.76455212898924814557707386119, −3.02723988755840065197137322263, −2.21964785285620202033777801894, −1.86598458648237018427246833584, −0.07796839221838006038521282421, 0.07796839221838006038521282421, 1.86598458648237018427246833584, 2.21964785285620202033777801894, 3.02723988755840065197137322263, 3.76455212898924814557707386119, 4.32778382847608198057906378047, 4.72510502641948934612071710553, 5.85183929586760904875550215167, 6.16462809648141667525249024065, 6.61915776285840484272313761167, 6.99323642123612158462826833226, 7.59633877458315899320980396190, 8.223913489044351500887777074324, 8.476978157658496165256371944774, 8.764627998005545138459719887879, 9.551628077936617551967454490951, 9.792468513161543745655948759513, 10.57600315855219390884594685293, 10.87219723725438341815567761614, 11.24695729373639647678619778021

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