Properties

Label 4-150e2-1.1-c1e2-0-2
Degree $4$
Conductor $22500$
Sign $1$
Analytic cond. $1.43461$
Root an. cond. $1.09442$
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  + 4-s + 9-s + 4·11-s + 16-s − 16·31-s + 36-s + 4·41-s + 4·44-s − 10·49-s + 20·59-s + 4·61-s + 64-s + 24·71-s + 81-s − 20·89-s + 4·99-s − 16·101-s + 20·109-s − 10·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s + 1.20·11-s + 1/4·16-s − 2.87·31-s + 1/6·36-s + 0.624·41-s + 0.603·44-s − 1.42·49-s + 2.60·59-s + 0.512·61-s + 1/8·64-s + 2.84·71-s + 1/9·81-s − 2.11·89-s + 0.402·99-s − 1.59·101-s + 1.91·109-s − 0.909·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/12·144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(22500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.43461\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 22500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.399903007\)
\(L(\frac12)\) \(\approx\) \(1.399903007\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−11.12199065028595017208430791111, −10.04002787682700509865642496375, −9.868458945526617237244084557500, −9.040702798039804053791802466769, −8.831748546434365307733167695606, −7.907415924764968545249330860332, −7.47947512213883760565427163251, −6.64959962043719359186062300669, −6.60420630283526755468874076032, −5.54667504701134155415335928612, −5.13544146251726925083565635926, −3.85245454580381340504020564721, −3.79143046769365773246334925917, −2.45815773509881651106722995929, −1.48708143658565903074344661320, 1.48708143658565903074344661320, 2.45815773509881651106722995929, 3.79143046769365773246334925917, 3.85245454580381340504020564721, 5.13544146251726925083565635926, 5.54667504701134155415335928612, 6.60420630283526755468874076032, 6.64959962043719359186062300669, 7.47947512213883760565427163251, 7.907415924764968545249330860332, 8.831748546434365307733167695606, 9.040702798039804053791802466769, 9.868458945526617237244084557500, 10.04002787682700509865642496375, 11.12199065028595017208430791111

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