Properties

Label 8-1200e4-1.1-c2e4-0-21
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $1.14304\times 10^{6}$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 8·7-s + 4·9-s + 40·13-s − 32·19-s + 32·21-s + 4·27-s − 32·31-s + 88·37-s + 160·39-s + 56·43-s + 24·49-s − 128·57-s − 64·61-s + 32·63-s − 328·67-s − 200·73-s + 112·79-s + 31·81-s + 320·91-s − 128·93-s − 296·97-s + 296·103-s + 80·109-s + 352·111-s + 160·117-s + 340·121-s + ⋯
L(s)  = 1  + 4/3·3-s + 8/7·7-s + 4/9·9-s + 3.07·13-s − 1.68·19-s + 1.52·21-s + 4/27·27-s − 1.03·31-s + 2.37·37-s + 4.10·39-s + 1.30·43-s + 0.489·49-s − 2.24·57-s − 1.04·61-s + 0.507·63-s − 4.89·67-s − 2.73·73-s + 1.41·79-s + 0.382·81-s + 3.51·91-s − 1.37·93-s − 3.05·97-s + 2.87·103-s + 0.733·109-s + 3.17·111-s + 1.36·117-s + 2.80·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.14304\times 10^{6}\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(8.109971645\)
\(L(\frac12)\) \(\approx\) \(8.109971645\)
\(L(2)\) 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$C_2^2$ \( 1 - 4 T + 4 p T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
5 \( 1 \)
good7$D_{4}$ \( ( 1 - 4 T + 12 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{4} \)
17$D_4\times C_2$ \( 1 - 220 T^{2} - 28218 T^{4} - 220 p^{4} T^{6} + p^{8} T^{8} \)
19$D_{4}$ \( ( 1 + 16 T + 426 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 1720 T^{2} + 1286322 T^{4} - 1720 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{4} \)
37$D_{4}$ \( ( 1 - 44 T + 1782 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 4060 T^{2} + 8942982 T^{4} - 4060 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 - 28 T + 3084 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 1064 T^{2} + 1942386 T^{4} + 1064 p^{4} T^{6} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 10300 T^{2} + 42096102 T^{4} - 10300 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 7300 T^{2} + 30092262 T^{4} - 7300 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 32 T + 6258 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 164 T + 14892 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 15124 T^{2} + 102823206 T^{4} - 15124 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 100 T + 11718 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 56 T + 12906 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 26872 T^{2} + 275326098 T^{4} - 26872 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 27940 T^{2} + 317327622 T^{4} - 27940 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 + 148 T + 22854 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.89451773521335461913529429910, −6.33145375050248219925412862348, −6.14857954264152358134398072830, −6.13760809400984208336137914541, −5.97529433200880994999514172575, −5.95419944094831362308471556727, −5.32652054354825111880838618346, −5.32399808311488612586998895021, −4.79437497648897878521946018352, −4.57066742864199618487174257627, −4.41396993753707649611116773356, −4.24904313521420040478495399251, −3.88032969197064023362818103990, −3.71196460909557438232381425155, −3.69922229334494770067822493574, −2.93351135935764387820976155290, −2.88768594277035593343612345804, −2.87899392767149723104521384033, −2.49292761033474519507191008391, −1.85793307687331360402134432633, −1.70214979276378549003362460116, −1.63908055503996504223069920118, −1.17335169961571792350480805234, −0.846778873332706800236466608443, −0.30629996211970745164458111073, 0.30629996211970745164458111073, 0.846778873332706800236466608443, 1.17335169961571792350480805234, 1.63908055503996504223069920118, 1.70214979276378549003362460116, 1.85793307687331360402134432633, 2.49292761033474519507191008391, 2.87899392767149723104521384033, 2.88768594277035593343612345804, 2.93351135935764387820976155290, 3.69922229334494770067822493574, 3.71196460909557438232381425155, 3.88032969197064023362818103990, 4.24904313521420040478495399251, 4.41396993753707649611116773356, 4.57066742864199618487174257627, 4.79437497648897878521946018352, 5.32399808311488612586998895021, 5.32652054354825111880838618346, 5.95419944094831362308471556727, 5.97529433200880994999514172575, 6.13760809400984208336137914541, 6.14857954264152358134398072830, 6.33145375050248219925412862348, 6.89451773521335461913529429910

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