Properties

Label 4-60e2-1.1-c3e2-0-0
Degree $4$
Conductor $3600$
Sign $1$
Analytic cond. $12.5324$
Root an. cond. $1.88151$
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  − 20·5-s − 9·9-s − 28·11-s + 240·19-s + 275·25-s − 192·29-s + 368·31-s + 260·41-s + 180·45-s + 202·49-s + 560·55-s − 532·59-s − 1.67e3·61-s + 2.04e3·71-s + 96·79-s + 81·81-s + 1.30e3·89-s − 4.80e3·95-s + 252·99-s + 3.37e3·101-s − 644·109-s − 2.07e3·121-s − 3.00e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s − 0.767·11-s + 2.89·19-s + 11/5·25-s − 1.22·29-s + 2.13·31-s + 0.990·41-s + 0.596·45-s + 0.588·49-s + 1.37·55-s − 1.17·59-s − 3.51·61-s + 3.40·71-s + 0.136·79-s + 1/9·81-s + 1.54·89-s − 5.18·95-s + 0.255·99-s + 3.32·101-s − 0.565·109-s − 1.55·121-s − 2.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(12.5324\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3600,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.054094314\)
\(L(\frac12)\) \(\approx\) \(1.054094314\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3$C_2$ \( 1 + p^{2} T^{2} \)
5$C_2$ \( 1 + 4 p T + p^{3} T^{2} \)
good7$C_2^2$ \( 1 - 202 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 14 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3494 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 5982 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 120 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 11010 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 96 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 184 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 63530 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 130 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 137110 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 6942 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 126358 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 266 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 838 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 540022 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 1020 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 543778 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 48 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 843270 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 650 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 594 T + p^{3} T^{2} )( 1 + 594 T + p^{3} T^{2} ) \)
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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

−15.18164340988960176128298056831, −14.11827393954947031647812105566, −13.96003666864334054483352860161, −13.19361116250847924129284430140, −12.40581078479412414450720146950, −11.96626871538909193857831557204, −11.59579945449408732215424199311, −11.00645567177374432889388036053, −10.43190617814385633843545324693, −9.508482528601366239430836201492, −9.045254492107358451481695548043, −7.936124825875806050052418129540, −7.81023775929059307732008724150, −7.30238410986779681023005762665, −6.23463771150623756282406870867, −5.23962165145979374907026878727, −4.61418603474678512847924090039, −3.52206508037291214315350357922, −2.92707269021384434319322824921, −0.75279567343431465149817209523, 0.75279567343431465149817209523, 2.92707269021384434319322824921, 3.52206508037291214315350357922, 4.61418603474678512847924090039, 5.23962165145979374907026878727, 6.23463771150623756282406870867, 7.30238410986779681023005762665, 7.81023775929059307732008724150, 7.936124825875806050052418129540, 9.045254492107358451481695548043, 9.508482528601366239430836201492, 10.43190617814385633843545324693, 11.00645567177374432889388036053, 11.59579945449408732215424199311, 11.96626871538909193857831557204, 12.40581078479412414450720146950, 13.19361116250847924129284430140, 13.96003666864334054483352860161, 14.11827393954947031647812105566, 15.18164340988960176128298056831

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