Properties

Label 4-252e2-1.1-c3e2-0-0
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  − 5·2-s + 17·4-s − 45·8-s + 89·16-s + 250·25-s − 332·29-s − 85·32-s − 900·37-s − 343·49-s − 1.25e3·50-s − 1.18e3·53-s + 1.66e3·58-s − 287·64-s + 4.50e3·74-s + 1.71e3·98-s + 4.25e3·100-s + 5.90e3·106-s + 108·109-s + 1.34e3·113-s − 5.64e3·116-s + 1.96e3·121-s + 127-s + 2.11e3·128-s + 131-s + 137-s + 139-s − 1.53e4·148-s + ⋯
L(s)  = 1  − 1.76·2-s + 17/8·4-s − 1.98·8-s + 1.39·16-s + 2·25-s − 2.12·29-s − 0.469·32-s − 3.99·37-s − 49-s − 3.53·50-s − 3.05·53-s + 3.75·58-s − 0.560·64-s + 7.06·74-s + 1.76·98-s + 17/4·100-s + 5.40·106-s + 0.0949·109-s + 1.11·113-s − 4.51·116-s + 1.47·121-s + 0.000698·127-s + 1.46·128-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 8.49·148-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

\(L(2)\) \(\approx\) \(0.4536332152\)
\(L(\frac12)\) \(\approx\) \(0.4536332152\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + 5 T + p^{3} T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + p^{3} T^{2} \)
good5$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 68 T + p^{3} T^{2} )( 1 + 68 T + p^{3} T^{2} ) \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
23$C_2$ \( ( 1 - 40 T + p^{3} T^{2} )( 1 + 40 T + p^{3} T^{2} ) \)
29$C_2$ \( ( 1 + 166 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 450 T + p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 180 T + p^{3} T^{2} )( 1 + 180 T + p^{3} T^{2} ) \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 + 590 T + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 740 T + p^{3} T^{2} )( 1 + 740 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 - 688 T + p^{3} T^{2} )( 1 + 688 T + p^{3} T^{2} ) \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 1384 T + p^{3} T^{2} )( 1 + 1384 T + p^{3} T^{2} ) \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
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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

−11.63250845780599637163960697604, −11.05492851383318895144459211919, −10.78700854169489903459336827058, −10.46430477521529128115785484782, −9.797904296815373446761075851128, −9.353687475414259363274825069655, −9.028433412611045027406713014986, −8.392659211479266124667662446050, −8.227148943344590358752722757210, −7.30924591156491555564271890825, −7.14949168286708015268952232180, −6.61999438487460971597940687937, −5.97596687320557144716823733695, −5.20910868986212743623347318663, −4.68731400372391173328406912804, −3.45945372621218328865467890163, −3.10933723748747777874024968018, −1.91420099927968933764265093089, −1.55479683112372824848673002048, −0.35574941328729113819783562134, 0.35574941328729113819783562134, 1.55479683112372824848673002048, 1.91420099927968933764265093089, 3.10933723748747777874024968018, 3.45945372621218328865467890163, 4.68731400372391173328406912804, 5.20910868986212743623347318663, 5.97596687320557144716823733695, 6.61999438487460971597940687937, 7.14949168286708015268952232180, 7.30924591156491555564271890825, 8.227148943344590358752722757210, 8.392659211479266124667662446050, 9.028433412611045027406713014986, 9.353687475414259363274825069655, 9.797904296815373446761075851128, 10.46430477521529128115785484782, 10.78700854169489903459336827058, 11.05492851383318895144459211919, 11.63250845780599637163960697604

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