Properties

Label 8-384e4-1.1-c5e4-0-1
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $1.43868\times 10^{7}$
Root an. cond. $7.84776$
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  − 486·9-s − 1.05e4·25-s + 4.99e4·49-s − 3.62e5·73-s + 1.77e5·81-s + 3.60e5·97-s + 4.54e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.48e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 5.13e6·225-s + ⋯
L(s)  = 1  − 2·9-s − 3.38·25-s + 2.96·49-s − 7.96·73-s + 3·81-s + 3.89·97-s + 2.81·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 6.76·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.43868\times 10^{7}\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.5175919289\)
\(L(\frac12)\) \(\approx\) \(0.5175919289\)
\(L(\frac{7}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 5282 T^{2} + p^{10} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 242 T + p^{5} T^{2} )^{2}( 1 + 242 T + p^{5} T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - 227050 T^{2} + p^{10} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
17$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
19$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
23$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 36304750 T^{2} + p^{10} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 6950 T + p^{5} T^{2} )^{2}( 1 + 6950 T + p^{5} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
41$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
43$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
47$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 617985986 T^{2} + p^{10} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 588563050 T^{2} + p^{10} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
67$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
71$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
73$C_2$ \( ( 1 + 90706 T + p^{5} T^{2} )^{4} \)
79$C_2$ \( ( 1 - 17050 T + p^{5} T^{2} )^{2}( 1 + 17050 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - 7877173786 T^{2} + p^{10} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 90242 T + p^{5} T^{2} )^{4} \)
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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

−7.54143170561936692896544448039, −7.08938762268203931113873380590, −7.00469677569301500266985583553, −6.56176616190729582609386688324, −6.11693313544474451237817664411, −5.92893912928341478483203293583, −5.82285711079206753399629935515, −5.67309889590733506713771895615, −5.62391088519168724538850193133, −5.07609379247902505498013551484, −4.68466801602957303722782278440, −4.35743311633985370324555631004, −4.31974550258758719856012797124, −3.83479861656503842065609789313, −3.63262264220150995925541377398, −3.20651424361689055606255754343, −2.88782858940028206805772487375, −2.87689873501416663878568012313, −2.20434661806228044682898258302, −2.04591865524768733433345063271, −1.87181747860969257149205965894, −1.29988851690560917590178695026, −0.856505901096000704017256424298, −0.43515881772642043979118623060, −0.12899905165008978575741692863, 0.12899905165008978575741692863, 0.43515881772642043979118623060, 0.856505901096000704017256424298, 1.29988851690560917590178695026, 1.87181747860969257149205965894, 2.04591865524768733433345063271, 2.20434661806228044682898258302, 2.87689873501416663878568012313, 2.88782858940028206805772487375, 3.20651424361689055606255754343, 3.63262264220150995925541377398, 3.83479861656503842065609789313, 4.31974550258758719856012797124, 4.35743311633985370324555631004, 4.68466801602957303722782278440, 5.07609379247902505498013551484, 5.62391088519168724538850193133, 5.67309889590733506713771895615, 5.82285711079206753399629935515, 5.92893912928341478483203293583, 6.11693313544474451237817664411, 6.56176616190729582609386688324, 7.00469677569301500266985583553, 7.08938762268203931113873380590, 7.54143170561936692896544448039

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