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Properties

Label 8-48e4-1.1-c15e4-0-1
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $2.20080\times 10^{7}$
Root an. cond. $8.27604$
Motivic weight $15$
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  − 7.65e6·9-s + 1.01e9·13-s + 1.10e11·25-s − 7.15e11·37-s + 1.22e13·49-s − 5.78e13·61-s + 2.67e14·73-s − 1.47e14·81-s − 1.50e15·97-s − 7.79e15·117-s − 5.03e15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.42e17·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 0.533·9-s + 4.49·13-s + 3.62·25-s − 1.23·37-s + 2.59·49-s − 2.35·61-s + 2.83·73-s − 0.715·81-s − 1.88·97-s − 2.40·117-s − 1.20·121-s + 8.64·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.20080\times 10^{7}\)
Root analytic conductor: \(8.27604\)
Motivic weight: \(15\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\)

Particular Values

\(L(8)\) \(\approx\) \(16.68841149\)
\(L(\frac12)\) \(\approx\) \(16.68841149\)
\(L(\frac{17}{2})\) 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$C_2^2$ \( 1 + 94550 p^{4} T^{2} + p^{30} T^{4} \)
good5$C_2^2$ \( ( 1 - 2209974346 p^{2} T^{2} + p^{30} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 125498449058 p^{2} T^{2} + p^{30} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 2515631802990118 T^{2} + p^{30} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 19572718 p T + p^{15} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 5179202542962652642 T^{2} + p^{30} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 30078947598155975594 T^{2} + p^{30} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + \)\(24\!\cdots\!10\)\( T^{2} + p^{30} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + \)\(97\!\cdots\!02\)\( T^{2} + p^{30} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - \)\(32\!\cdots\!86\)\( T^{2} + p^{30} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 178908455266 T + p^{15} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - \)\(28\!\cdots\!06\)\( T^{2} + p^{30} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - \)\(60\!\cdots\!70\)\( T^{2} + p^{30} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + \)\(22\!\cdots\!86\)\( T^{2} + p^{30} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + \)\(23\!\cdots\!70\)\( T^{2} + p^{30} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + \)\(73\!\cdots\!42\)\( T^{2} + p^{30} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 14469775333882 T + p^{15} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + \)\(27\!\cdots\!30\)\( T^{2} + p^{30} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + \)\(49\!\cdots\!58\)\( T^{2} + p^{30} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 66786962361658 T + p^{15} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - \)\(20\!\cdots\!54\)\( T^{2} + p^{30} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - \)\(48\!\cdots\!02\)\( T^{2} + p^{30} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - \)\(33\!\cdots\!94\)\( T^{2} + p^{30} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 375170793938030 T + p^{15} T^{2} )^{4} \)
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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

−8.496409790432716710401090487074, −8.424771371638327941002219731326, −8.093353391676571613274937956617, −7.33728038281080805144677567052, −7.21775543341091262270398651033, −6.79822096231824439824239780855, −6.33761478187108456681660432742, −6.23755635808091637694483423312, −6.12814976085332948935557865351, −5.43066098635030943280914414504, −5.31225815278596181963730444402, −5.04924366922779471289801340004, −4.33138464077359477669754764547, −4.06007254625631983897053258357, −3.93002501915916307720937856840, −3.35518160707461158345502777291, −3.14604389155724359526172036267, −2.95061526568260118938856646253, −2.53386077561402317393481739727, −1.79862510225295728931495285638, −1.49427906368569442329413678384, −1.43009735753530188385247903199, −0.74614533500571646368450321670, −0.69318451926498322943669799507, −0.56202054419060885065662716996, 0.56202054419060885065662716996, 0.69318451926498322943669799507, 0.74614533500571646368450321670, 1.43009735753530188385247903199, 1.49427906368569442329413678384, 1.79862510225295728931495285638, 2.53386077561402317393481739727, 2.95061526568260118938856646253, 3.14604389155724359526172036267, 3.35518160707461158345502777291, 3.93002501915916307720937856840, 4.06007254625631983897053258357, 4.33138464077359477669754764547, 5.04924366922779471289801340004, 5.31225815278596181963730444402, 5.43066098635030943280914414504, 6.12814976085332948935557865351, 6.23755635808091637694483423312, 6.33761478187108456681660432742, 6.79822096231824439824239780855, 7.21775543341091262270398651033, 7.33728038281080805144677567052, 8.093353391676571613274937956617, 8.424771371638327941002219731326, 8.496409790432716710401090487074

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