Properties

Label 8-48e4-1.1-c4e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $606.097$
Root an. cond. $2.22750$
Motivic weight $4$
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  + 4·3-s − 24·7-s + 58·9-s − 248·13-s − 824·19-s − 96·21-s + 228·25-s + 724·27-s − 3.80e3·31-s − 2.80e3·37-s − 992·39-s + 968·43-s − 5.50e3·49-s − 3.29e3·57-s + 8.58e3·61-s − 1.39e3·63-s + 1.51e4·67-s + 5.51e3·73-s + 912·75-s + 1.06e3·79-s + 67·81-s + 5.95e3·91-s − 1.52e4·93-s + 3.24e4·97-s − 3.44e4·103-s − 632·109-s − 1.12e4·111-s + ⋯
L(s)  = 1  + 4/9·3-s − 0.489·7-s + 0.716·9-s − 1.46·13-s − 2.28·19-s − 0.217·21-s + 0.364·25-s + 0.993·27-s − 3.95·31-s − 2.05·37-s − 0.652·39-s + 0.523·43-s − 2.29·49-s − 1.01·57-s + 2.30·61-s − 0.350·63-s + 3.36·67-s + 1.03·73-s + 0.162·75-s + 0.170·79-s + 0.0102·81-s + 0.718·91-s − 1.75·93-s + 3.44·97-s − 3.24·103-s − 0.0531·109-s − 0.911·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(606.097\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5360281225\)
\(L(\frac12)\) \(\approx\) \(0.5360281225\)
\(L(3)\) 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$C_2^2$ \( 1 - 4 T - 14 p T^{2} - 4 p^{4} T^{3} + p^{8} T^{4} \)
good5$D_4\times C_2$ \( 1 - 228 T^{2} + 45446 T^{4} - 228 p^{8} T^{6} + p^{16} T^{8} \)
7$D_{4}$ \( ( 1 + 12 T + 2966 T^{2} + 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 14820 T^{2} + 30482054 T^{4} - 14820 p^{8} T^{6} + p^{16} T^{8} \)
13$D_{4}$ \( ( 1 + 124 T + 31014 T^{2} + 124 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 211716 T^{2} + 22389387014 T^{4} - 211716 p^{8} T^{6} + p^{16} T^{8} \)
19$D_{4}$ \( ( 1 + 412 T + 301206 T^{2} + 412 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 634116 T^{2} + 213728847494 T^{4} - 634116 p^{8} T^{6} + p^{16} T^{8} \)
29$D_4\times C_2$ \( 1 - 2601316 T^{2} + 2691910499334 T^{4} - 2601316 p^{8} T^{6} + p^{16} T^{8} \)
31$D_{4}$ \( ( 1 + 1900 T + 2523030 T^{2} + 1900 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 1404 T + 2773478 T^{2} + 1404 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 4336900 T^{2} + 9972871940742 T^{4} - 4336900 p^{8} T^{6} + p^{16} T^{8} \)
43$D_{4}$ \( ( 1 - 484 T + 6804438 T^{2} - 484 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 18491140 T^{2} + 132894384489222 T^{4} - 18491140 p^{8} T^{6} + p^{16} T^{8} \)
53$D_4\times C_2$ \( 1 - 2855652 T^{2} - 56957106011002 T^{4} - 2855652 p^{8} T^{6} + p^{16} T^{8} \)
59$D_4\times C_2$ \( 1 - 33977316 T^{2} + 531124679369606 T^{4} - 33977316 p^{8} T^{6} + p^{16} T^{8} \)
61$D_{4}$ \( ( 1 - 4292 T + 470046 p T^{2} - 4292 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 7556 T + 54528726 T^{2} - 7556 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 72386052 T^{2} + 2410878712841606 T^{4} - 72386052 p^{8} T^{6} + p^{16} T^{8} \)
73$D_{4}$ \( ( 1 - 2756 T + 54382278 T^{2} - 2756 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 532 T + 41802006 T^{2} - 532 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 99356772 T^{2} + 6615483960569606 T^{4} - 99356772 p^{8} T^{6} + p^{16} T^{8} \)
89$D_4\times C_2$ \( 1 - 186937348 T^{2} + 16504528715125638 T^{4} - 186937348 p^{8} T^{6} + p^{16} T^{8} \)
97$D_{4}$ \( ( 1 - 16228 T + 224175558 T^{2} - 16228 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.84330390249714442441000052506, −10.59027246573750934093601213334, −10.37366436170507231714683233460, −9.725723068801479128936419073226, −9.629811636406627527339393698583, −9.568408310604394007115582192537, −8.710630804886119223757142491383, −8.675674166815932750244584319780, −8.570718314209767436795824291181, −7.83262603731394397823167542772, −7.46260031816399037906513065234, −7.29443601192892709898381682623, −6.78456308949997375857755287549, −6.46048965694804543777575201918, −6.36263046929839813078395454966, −5.25879470161811167342838583969, −5.25539272040424110299287063076, −4.95980456104048854719918554999, −4.11000853571574819028106162533, −3.70164229403187575958540404362, −3.55504781537720858176141800861, −2.46238957844519133733820762097, −2.22450807919906791099545897465, −1.58444309195845848303725559886, −0.22054334635729659442557635091, 0.22054334635729659442557635091, 1.58444309195845848303725559886, 2.22450807919906791099545897465, 2.46238957844519133733820762097, 3.55504781537720858176141800861, 3.70164229403187575958540404362, 4.11000853571574819028106162533, 4.95980456104048854719918554999, 5.25539272040424110299287063076, 5.25879470161811167342838583969, 6.36263046929839813078395454966, 6.46048965694804543777575201918, 6.78456308949997375857755287549, 7.29443601192892709898381682623, 7.46260031816399037906513065234, 7.83262603731394397823167542772, 8.570718314209767436795824291181, 8.675674166815932750244584319780, 8.710630804886119223757142491383, 9.568408310604394007115582192537, 9.629811636406627527339393698583, 9.725723068801479128936419073226, 10.37366436170507231714683233460, 10.59027246573750934093601213334, 10.84330390249714442441000052506

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