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Properties

Label 4-735e2-1.1-c0e2-0-1
Degree $4$
Conductor $540225$
Sign $1$
Analytic cond. $0.134551$
Root an. cond. $0.605650$
Motivic weight $0$
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  − 2·3-s + 2·5-s + 3·9-s − 4·15-s − 16-s + 3·25-s − 4·27-s + 6·45-s + 2·48-s − 6·75-s − 2·80-s + 5·81-s − 4·109-s + 2·121-s + 4·125-s + 127-s + 131-s − 8·135-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯
L(s)  = 1  − 2·3-s + 2·5-s + 3·9-s − 4·15-s − 16-s + 3·25-s − 4·27-s + 6·45-s + 2·48-s − 6·75-s − 2·80-s + 5·81-s − 4·109-s + 2·121-s + 4·125-s + 127-s + 131-s − 8·135-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.134551\)
Root analytic conductor: \(0.605650\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 540225,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6787230964\)
\(L(\frac12)\) \(\approx\) \(0.6787230964\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.84450122207702245493234375661, −10.48767679781548031122155624154, −9.890099789933889773189314341736, −9.816805548434013102121918236944, −9.190294197280203749628783938272, −9.032920328246445280370002573946, −8.220174536731727986824345885605, −7.58400158631477052659106977324, −6.97633822972332313412119909893, −6.59345587307017599841416552965, −6.50765901467596082400325519957, −5.83921999502597708888875756938, −5.39579545452888574929467137370, −5.33204746641583071004976068193, −4.43411112815052962302483509080, −4.41019616365952847647548395155, −3.28390921012292782619901431060, −2.36507033203502314396116263133, −1.82901295326634746793103000389, −1.11613813786054256982614528162, 1.11613813786054256982614528162, 1.82901295326634746793103000389, 2.36507033203502314396116263133, 3.28390921012292782619901431060, 4.41019616365952847647548395155, 4.43411112815052962302483509080, 5.33204746641583071004976068193, 5.39579545452888574929467137370, 5.83921999502597708888875756938, 6.50765901467596082400325519957, 6.59345587307017599841416552965, 6.97633822972332313412119909893, 7.58400158631477052659106977324, 8.220174536731727986824345885605, 9.032920328246445280370002573946, 9.190294197280203749628783938272, 9.816805548434013102121918236944, 9.890099789933889773189314341736, 10.48767679781548031122155624154, 10.84450122207702245493234375661

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