Properties

Label 4-350e2-1.1-c3e2-0-5
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $426.450$
Root an. cond. $4.54430$
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  − 4·4-s + 50·9-s − 54·11-s + 16·16-s − 124·19-s − 282·29-s − 248·31-s − 200·36-s − 708·41-s + 216·44-s − 49·49-s − 696·59-s + 820·61-s − 64·64-s − 678·71-s + 496·76-s − 1.46e3·79-s + 1.77e3·81-s − 1.92e3·89-s − 2.70e3·99-s + 2.94e3·101-s + 998·109-s + 1.12e3·116-s − 475·121-s + 992·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.85·9-s − 1.48·11-s + 1/4·16-s − 1.49·19-s − 1.80·29-s − 1.43·31-s − 0.925·36-s − 2.69·41-s + 0.740·44-s − 1/7·49-s − 1.53·59-s + 1.72·61-s − 1/8·64-s − 1.13·71-s + 0.748·76-s − 2.08·79-s + 2.42·81-s − 2.28·89-s − 2.74·99-s + 2.89·101-s + 0.876·109-s + 0.902·116-s − 0.356·121-s + 0.718·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(426.450\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 122500,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7372200973\)
\(L(\frac12)\) \(\approx\) \(0.7372200973\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + p^{2} T^{2} \)
good3$C_2^2$ \( 1 - 50 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 27 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 298 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 9250 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 62 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 13309 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 141 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 91415 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 354 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 114493 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 197242 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 204118 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 348 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 410 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 479725 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 339 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 773134 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 731 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 864790 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 960 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 29746 T^{2} + p^{6} T^{4} \)
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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

−11.52965996470426884705084849394, −10.54482361091870062466469265850, −10.32875759299709447277402045233, −10.10892822430006430155690565664, −9.497689087696767393107919467453, −8.940200409973147858371027807964, −8.577124298444843599746367814961, −7.82892198967473867272153589300, −7.65176052445653907509547879938, −6.92095039226233986386859747958, −6.75730210864393543489179777664, −5.72890474464308201113530567448, −5.43497261635893412778532237436, −4.72818675987604307057435996847, −4.31131491497541652775909266376, −3.73958255647490258788318272966, −3.06716999397876975228221872792, −1.90859871342238872511482614946, −1.71622393891674759122324223518, −0.29432559864567843462942388783, 0.29432559864567843462942388783, 1.71622393891674759122324223518, 1.90859871342238872511482614946, 3.06716999397876975228221872792, 3.73958255647490258788318272966, 4.31131491497541652775909266376, 4.72818675987604307057435996847, 5.43497261635893412778532237436, 5.72890474464308201113530567448, 6.75730210864393543489179777664, 6.92095039226233986386859747958, 7.65176052445653907509547879938, 7.82892198967473867272153589300, 8.577124298444843599746367814961, 8.940200409973147858371027807964, 9.497689087696767393107919467453, 10.10892822430006430155690565664, 10.32875759299709447277402045233, 10.54482361091870062466469265850, 11.52965996470426884705084849394

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