Properties

Label 8-1350e4-1.1-c3e4-0-7
Degree $8$
Conductor $3.322\times 10^{12}$
Sign $1$
Analytic cond. $4.02531\times 10^{7}$
Root an. cond. $8.92482$
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  − 8·4-s + 60·11-s + 48·16-s − 92·19-s − 180·29-s − 40·31-s + 600·41-s − 480·44-s + 944·49-s − 180·59-s + 272·61-s − 256·64-s + 1.56e3·71-s + 736·76-s + 892·79-s − 1.14e3·89-s + 1.26e3·101-s + 2.12e3·109-s + 1.44e3·116-s + 2.97e3·121-s + 320·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s + 1.64·11-s + 3/4·16-s − 1.11·19-s − 1.15·29-s − 0.231·31-s + 2.28·41-s − 1.64·44-s + 2.75·49-s − 0.397·59-s + 0.570·61-s − 1/2·64-s + 2.60·71-s + 1.11·76-s + 1.27·79-s − 1.35·89-s + 1.24·101-s + 1.86·109-s + 1.15·116-s + 2.23·121-s + 0.231·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{12} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(4.02531\times 10^{7}\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{12} \cdot 5^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(9.082103436\)
\(L(\frac12)\) \(\approx\) \(9.082103436\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$D_4\times C_2$ \( 1 - 944 T^{2} + 439182 T^{4} - 944 p^{6} T^{6} + p^{12} T^{8} \)
11$D_{4}$ \( ( 1 - 30 T - 137 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 7610 T^{2} + 23829243 T^{4} - 7610 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 - 9752 T^{2} + 67798014 T^{4} - 9752 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 + 46 T + 9522 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 38930 T^{2} + 672236403 T^{4} - 38930 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 90 T + 41542 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 20 T - 15918 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 198410 T^{2} + 14970363243 T^{4} - 198410 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 300 T + 133126 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 122528 T^{2} + 14637831294 T^{4} - 122528 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 211154 T^{2} + 30865131987 T^{4} - 211154 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 555800 T^{2} + 121271194158 T^{4} - 555800 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 + 90 T + 285019 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 136 T + 75861 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 206384 T^{2} + 161986834302 T^{4} - 206384 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 780 T + 825397 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1333436 T^{2} + 738025881702 T^{4} - 1333436 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 - 446 T + 993282 T^{2} - 446 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 429116 T^{2} + 631111112502 T^{4} - 429116 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 570 T + 591334 T^{2} + 570 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 633914 T^{2} + 1060829059707 T^{4} - 633914 p^{6} T^{6} + p^{12} T^{8} \)
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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

−6.71594629836462909346436545208, −6.14831351364506833963974507954, −5.83126510628510450262138474726, −5.80118817602286733965974825734, −5.77286007552757512503045133532, −5.38234200662765772245000296685, −5.23097329140089032257859917853, −4.81009413969813071498768018243, −4.53394640429396755293876339230, −4.41446214359072984904800509784, −4.15369681185355717172806758290, −4.01163760632579143390317417631, −3.86634633623179157698862000198, −3.65201077948657106865614951706, −3.09187477545916434665116449814, −3.02598939811459535024504032122, −2.90431468810883839304620062591, −2.20515794907016059230167773112, −1.93176105637410536321590256876, −1.82616345797047891115611585626, −1.82110501955969540507641547745, −0.870181007771606726705608329100, −0.71987631069724863626427860503, −0.68978949508392375918008140502, −0.45635711070082137111523676152, 0.45635711070082137111523676152, 0.68978949508392375918008140502, 0.71987631069724863626427860503, 0.870181007771606726705608329100, 1.82110501955969540507641547745, 1.82616345797047891115611585626, 1.93176105637410536321590256876, 2.20515794907016059230167773112, 2.90431468810883839304620062591, 3.02598939811459535024504032122, 3.09187477545916434665116449814, 3.65201077948657106865614951706, 3.86634633623179157698862000198, 4.01163760632579143390317417631, 4.15369681185355717172806758290, 4.41446214359072984904800509784, 4.53394640429396755293876339230, 4.81009413969813071498768018243, 5.23097329140089032257859917853, 5.38234200662765772245000296685, 5.77286007552757512503045133532, 5.80118817602286733965974825734, 5.83126510628510450262138474726, 6.14831351364506833963974507954, 6.71594629836462909346436545208

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