Properties

Label 4-252e2-1.1-c3e2-0-3
Degree $4$
Conductor $63504$
Sign $1$
Analytic cond. $221.071$
Root an. cond. $3.85596$
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  + 14·7-s + 52·13-s + 136·19-s + 2·25-s + 424·31-s + 436·37-s + 520·43-s + 147·49-s − 644·61-s + 712·67-s − 452·73-s + 880·79-s + 728·91-s − 2.66e3·97-s − 3.60e3·103-s − 908·109-s − 2.41e3·121-s + 127-s + 131-s + 1.90e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.10·13-s + 1.64·19-s + 0.0159·25-s + 2.45·31-s + 1.93·37-s + 1.84·43-s + 3/7·49-s − 1.35·61-s + 1.29·67-s − 0.724·73-s + 1.25·79-s + 0.838·91-s − 2.78·97-s − 3.45·103-s − 0.797·109-s − 1.81·121-s + 0.000698·127-s + 0.000666·131-s + 1.24·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63504\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(221.071\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 63504,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.552817047\)
\(L(\frac12)\) \(\approx\) \(3.552817047\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p T )^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 2410 T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3526 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 68 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 22066 T^{2} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 15734 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 212 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 218 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 19658 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 260 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 37294 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 70954 T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 329110 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 + 322 T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 356 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 110 p^{2} T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 + 226 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 440 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 1079062 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 1367350 T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \)
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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

−11.79555475117144036778369935112, −11.42086147646382569505623768644, −10.76979321619854562321758876675, −10.69659792131246063523854487557, −9.777914962303618750456575796840, −9.489088443346680409681878927088, −9.021465807306124843412947455783, −8.220364954879789577485401594375, −7.988878497066362385053585269349, −7.61555064592006891886496693167, −6.73492877581426153111022079794, −6.36863771726325589793122002884, −5.61362793115159714422014026867, −5.28697865737650297740388906317, −4.31248657651245955722318739806, −4.14590453978366792251602708723, −3.03127391329181947439969608492, −2.58146475321503156828531832470, −1.31686503132134081318149149323, −0.892460338966400222661375271724, 0.892460338966400222661375271724, 1.31686503132134081318149149323, 2.58146475321503156828531832470, 3.03127391329181947439969608492, 4.14590453978366792251602708723, 4.31248657651245955722318739806, 5.28697865737650297740388906317, 5.61362793115159714422014026867, 6.36863771726325589793122002884, 6.73492877581426153111022079794, 7.61555064592006891886496693167, 7.988878497066362385053585269349, 8.220364954879789577485401594375, 9.021465807306124843412947455783, 9.489088443346680409681878927088, 9.777914962303618750456575796840, 10.69659792131246063523854487557, 10.76979321619854562321758876675, 11.42086147646382569505623768644, 11.79555475117144036778369935112

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