Properties

Label 8-384e4-1.1-c3e4-0-5
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $263505.$
Root an. cond. $4.75990$
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  − 18·9-s − 136·17-s + 484·25-s + 104·41-s − 972·49-s − 1.35e3·73-s + 243·81-s − 936·89-s − 712·97-s + 5.51e3·113-s + 4.52e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.44e3·153-s + 157-s + 163-s + 167-s + 5.65e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s − 1.94·17-s + 3.87·25-s + 0.396·41-s − 2.83·49-s − 2.16·73-s + 1/3·81-s − 1.11·89-s − 0.745·97-s + 4.58·113-s + 3.39·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 1.29·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.57·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(263505.\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.732480290\)
\(L(\frac12)\) \(\approx\) \(1.732480290\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 242 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 486 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2262 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 2826 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 2 p T + p^{3} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 11014 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 20462 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 8450 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 47414 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 27578 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 26 T + p^{3} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 95510 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 88574 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 166894 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 278262 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 86838 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 207142 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 604430 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 338 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 363350 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 70278 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 234 T + p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 178 T + p^{3} T^{2} )^{4} \)
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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

−7.899503691105174176646368595144, −7.40226151026692747736564609205, −7.23753796132891260030602296272, −6.92835586582445081438932703400, −6.83248560405468646779664952422, −6.48318552662670867163721831558, −6.40113102970951909394524857088, −5.91478345948786207914828955378, −5.85020192010675186257053136414, −5.49283355215879112799614352490, −4.88424119895091956135116504667, −4.87793183203197561233041376106, −4.69010627665298541652862564958, −4.54507972618124683788052805180, −4.08496260060414511643724764386, −3.55137003746182330282803491242, −3.40875374196671540546513024122, −2.93015530507394179874910201207, −2.83489369265530440013514325255, −2.42598263830314613612738146260, −2.11915038568140354187990086097, −1.41718853309754787593893911810, −1.35427127267376563876911138928, −0.64700293773732759806390948661, −0.25486677170884438806167587978, 0.25486677170884438806167587978, 0.64700293773732759806390948661, 1.35427127267376563876911138928, 1.41718853309754787593893911810, 2.11915038568140354187990086097, 2.42598263830314613612738146260, 2.83489369265530440013514325255, 2.93015530507394179874910201207, 3.40875374196671540546513024122, 3.55137003746182330282803491242, 4.08496260060414511643724764386, 4.54507972618124683788052805180, 4.69010627665298541652862564958, 4.87793183203197561233041376106, 4.88424119895091956135116504667, 5.49283355215879112799614352490, 5.85020192010675186257053136414, 5.91478345948786207914828955378, 6.40113102970951909394524857088, 6.48318552662670867163721831558, 6.83248560405468646779664952422, 6.92835586582445081438932703400, 7.23753796132891260030602296272, 7.40226151026692747736564609205, 7.899503691105174176646368595144

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