Properties

Label 4-210e2-1.1-c5e2-0-0
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $1134.38$
Root an. cond. $5.80349$
Motivic weight $5$
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  − 4·2-s + 9·3-s + 25·5-s − 36·6-s + 119·7-s + 64·8-s − 100·10-s − 720·11-s + 2.23e3·13-s − 476·14-s + 225·15-s − 256·16-s − 654·17-s − 1.64e3·19-s + 1.07e3·21-s + 2.88e3·22-s + 3.12e3·23-s + 576·24-s − 8.92e3·26-s − 729·27-s + 1.00e4·29-s − 900·30-s − 8.18e3·31-s − 6.48e3·33-s + 2.61e3·34-s + 2.97e3·35-s + 7.29e3·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.447·5-s − 0.408·6-s + 0.917·7-s + 0.353·8-s − 0.316·10-s − 1.79·11-s + 3.65·13-s − 0.649·14-s + 0.258·15-s − 1/4·16-s − 0.548·17-s − 1.04·19-s + 0.529·21-s + 1.26·22-s + 1.23·23-s + 0.204·24-s − 2.58·26-s − 0.192·27-s + 2.21·29-s − 0.182·30-s − 1.52·31-s − 1.03·33-s + 0.388·34-s + 0.410·35-s + 0.875·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1134.38\)
Root analytic conductor: \(5.80349\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.854066591\)
\(L(\frac12)\) \(\approx\) \(3.854066591\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{2} T + p^{4} T^{2} \)
3$C_2$ \( 1 - p^{2} T + p^{4} T^{2} \)
5$C_2$ \( 1 - p^{2} T + p^{4} T^{2} \)
7$C_2$ \( 1 - 17 p T + p^{5} T^{2} \)
good11$C_2^2$ \( 1 + 720 T + 357349 T^{2} + 720 p^{5} T^{3} + p^{10} T^{4} \)
13$C_2$ \( ( 1 - 1115 T + p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 654 T - 992141 T^{2} + 654 p^{5} T^{3} + p^{10} T^{4} \)
19$C_2^2$ \( 1 + 1643 T + 223350 T^{2} + 1643 p^{5} T^{3} + p^{10} T^{4} \)
23$C_2^2$ \( 1 - 3126 T + 3335533 T^{2} - 3126 p^{5} T^{3} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 5010 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8183 T + 38332338 T^{2} + 8183 p^{5} T^{3} + p^{10} T^{4} \)
37$C_2^2$ \( 1 - 7291 T - 16185276 T^{2} - 7291 p^{5} T^{3} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 15420 T + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 11273 T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 3066 T - 219944651 T^{2} + 3066 p^{5} T^{3} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 25416 T + 227777563 T^{2} - 25416 p^{5} T^{3} + p^{10} T^{4} \)
59$C_2^2$ \( 1 + 10014 T - 614644103 T^{2} + 10014 p^{5} T^{3} + p^{10} T^{4} \)
61$C_2^2$ \( 1 + 34034 T + 313716855 T^{2} + 34034 p^{5} T^{3} + p^{10} T^{4} \)
67$C_2^2$ \( 1 - 42073 T + 420012222 T^{2} - 42073 p^{5} T^{3} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 17772 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 85525 T + 5241454032 T^{2} - 85525 p^{5} T^{3} + p^{10} T^{4} \)
79$C_2^2$ \( 1 - 93127 T + 5595581730 T^{2} - 93127 p^{5} T^{3} + p^{10} T^{4} \)
83$C_2$ \( ( 1 - 41172 T + p^{5} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 49254 T - 3158102933 T^{2} + 49254 p^{5} T^{3} + p^{10} T^{4} \)
97$C_2$ \( ( 1 + 119002 T + p^{5} T^{2} )^{2} \)
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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

−11.16084181062837326347271864600, −11.03194686488995354889461768105, −10.71478709617430258045036909518, −10.69705922468683485169023624526, −9.519000068474384752948185350954, −9.201909268349711315344615568552, −8.576643219186045777455738298669, −8.404943496831394648216412659455, −8.005528198018817834079904948546, −7.47750751278127284728376731974, −6.56958696323754635149182958184, −6.06894947316395010914161387173, −5.61444223482952167994179797567, −4.81089320644460318533984478236, −4.17929635930082874913760146499, −3.53985092726442070386163548348, −2.59984273533442238049647742205, −2.13350344647357023087147764541, −1.06624201991441182084479983330, −0.813068202847826686903537012011, 0.813068202847826686903537012011, 1.06624201991441182084479983330, 2.13350344647357023087147764541, 2.59984273533442238049647742205, 3.53985092726442070386163548348, 4.17929635930082874913760146499, 4.81089320644460318533984478236, 5.61444223482952167994179797567, 6.06894947316395010914161387173, 6.56958696323754635149182958184, 7.47750751278127284728376731974, 8.005528198018817834079904948546, 8.404943496831394648216412659455, 8.576643219186045777455738298669, 9.201909268349711315344615568552, 9.519000068474384752948185350954, 10.69705922468683485169023624526, 10.71478709617430258045036909518, 11.03194686488995354889461768105, 11.16084181062837326347271864600

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