Properties

Label 4-210e2-1.1-c1e2-0-2
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $2.81185$
Root an. cond. $1.29493$
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 − 2·5-s − 9-s − 4·11-s + 16-s + 4·19-s + 2·20-s − 25-s + 12·29-s + 12·31-s + 36-s + 12·41-s + 4·44-s + 2·45-s − 49-s + 8·55-s + 16·59-s + 20·61-s − 64-s − 28·71-s − 4·76-s − 8·79-s − 2·80-s + 81-s − 20·89-s − 8·95-s + 4·99-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 0.917·19-s + 0.447·20-s − 1/5·25-s + 2.22·29-s + 2.15·31-s + 1/6·36-s + 1.87·41-s + 0.603·44-s + 0.298·45-s − 1/7·49-s + 1.07·55-s + 2.08·59-s + 2.56·61-s − 1/8·64-s − 3.32·71-s − 0.458·76-s − 0.900·79-s − 0.223·80-s + 1/9·81-s − 2.11·89-s − 0.820·95-s + 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.81185\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8733835266\)
\(L(\frac12)\) \(\approx\) \(0.8733835266\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−12.85341074076681028669562334064, −11.87516824747386644714238969213, −11.77637777828850373241937052240, −11.36285221408295252654928188196, −10.55799777997487066299444878398, −10.05410291922253338571499584509, −9.976663410368504927093723199896, −9.049255686670733671635867707160, −8.534746866712033934298779236000, −8.015255240138304758518364371083, −7.896812855572944601763094997567, −7.05683691550879363480497476428, −6.56577000271706956530288027022, −5.62030931674211299426613422821, −5.34915356396605090891664883229, −4.34401055138544843684914808244, −4.24627869961818121660718616196, −2.96870758936338633814540348254, −2.70745549179855071060634531176, −0.841117477816508379795875374473, 0.841117477816508379795875374473, 2.70745549179855071060634531176, 2.96870758936338633814540348254, 4.24627869961818121660718616196, 4.34401055138544843684914808244, 5.34915356396605090891664883229, 5.62030931674211299426613422821, 6.56577000271706956530288027022, 7.05683691550879363480497476428, 7.896812855572944601763094997567, 8.015255240138304758518364371083, 8.534746866712033934298779236000, 9.049255686670733671635867707160, 9.976663410368504927093723199896, 10.05410291922253338571499584509, 10.55799777997487066299444878398, 11.36285221408295252654928188196, 11.77637777828850373241937052240, 11.87516824747386644714238969213, 12.85341074076681028669562334064

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