Properties

Label 8-1014e4-1.1-c3e4-0-0
Degree $8$
Conductor $1.057\times 10^{12}$
Sign $1$
Analytic cond. $1.28119\times 10^{7}$
Root an. cond. $7.73485$
Motivic weight $3$
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  + 12·3-s − 8·4-s + 90·9-s − 96·12-s + 48·16-s − 198·17-s + 28·23-s + 79·25-s + 540·27-s + 242·29-s − 720·36-s − 1.29e3·43-s + 576·48-s + 995·49-s − 2.37e3·51-s − 1.45e3·53-s − 1.69e3·61-s − 256·64-s + 1.58e3·68-s + 336·69-s + 948·75-s − 1.19e3·79-s + 2.83e3·81-s + 2.90e3·87-s − 224·92-s − 632·100-s + 902·101-s + ⋯
L(s)  = 1  + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s + 3/4·16-s − 2.82·17-s + 0.253·23-s + 0.631·25-s + 3.84·27-s + 1.54·29-s − 3.33·36-s − 4.57·43-s + 1.73·48-s + 2.90·49-s − 6.52·51-s − 3.77·53-s − 3.55·61-s − 1/2·64-s + 2.82·68-s + 0.586·69-s + 1.45·75-s − 1.70·79-s + 35/9·81-s + 3.57·87-s − 0.253·92-s − 0.631·100-s + 0.888·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(1.28119\times 10^{7}\)
Root analytic conductor: \(7.73485\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.164677471\)
\(L(\frac12)\) \(\approx\) \(1.164677471\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
13 \( 1 \)
good5$D_4\times C_2$ \( 1 - 79 T^{2} + 4376 T^{4} - 79 p^{6} T^{6} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 - 995 T^{2} + 469176 T^{4} - 995 p^{6} T^{6} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 296 p T^{2} + 5221694 T^{4} - 296 p^{7} T^{6} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 99 T + 10762 T^{2} + 99 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 5772 T^{2} + 99664150 T^{4} - 5772 p^{6} T^{6} + p^{12} T^{8} \)
23$D_{4}$ \( ( 1 - 14 T + 23710 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 121 T + 32080 T^{2} - 121 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 103907 T^{2} + 4416594168 T^{4} - 103907 p^{6} T^{6} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 126615 T^{2} + 9113832616 T^{4} - 126615 p^{6} T^{6} + p^{12} T^{8} \)
41$D_4\times C_2$ \( 1 - 71843 T^{2} + 2562792900 T^{4} - 71843 p^{6} T^{6} + p^{12} T^{8} \)
43$D_{4}$ \( ( 1 + 15 p T + 242662 T^{2} + 15 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 185704 T^{2} + 23586610862 T^{4} - 185704 p^{6} T^{6} + p^{12} T^{8} \)
53$D_{4}$ \( ( 1 + 729 T + 416986 T^{2} + 729 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 170872 T^{2} + 11613194750 T^{4} - 170872 p^{6} T^{6} + p^{12} T^{8} \)
61$D_{4}$ \( ( 1 + 846 T + 630199 T^{2} + 846 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 230099 T^{2} - 36991091568 T^{4} - 230099 p^{6} T^{6} + p^{12} T^{8} \)
71$D_4\times C_2$ \( 1 - 39880 T^{2} - 183563665906 T^{4} - 39880 p^{6} T^{6} + p^{12} T^{8} \)
73$D_4\times C_2$ \( 1 - 210650 T^{2} - 1979910597 T^{4} - 210650 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 + 599 T + 1037922 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 2044652 T^{2} + 1689597395286 T^{4} - 2044652 p^{6} T^{6} + p^{12} T^{8} \)
89$D_4\times C_2$ \( 1 - 2163632 T^{2} + 2127095528190 T^{4} - 2163632 p^{6} T^{6} + p^{12} T^{8} \)
97$D_4\times C_2$ \( 1 - 2925767 T^{2} + 3776322183024 T^{4} - 2925767 p^{6} T^{6} + p^{12} T^{8} \)
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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

−6.88879545586550197711788606706, −6.36030411698765390038075524672, −6.32074456781066123415077531416, −6.22490938668353801448659156404, −6.17449691995099878232322613664, −5.31321179394805222916478843598, −5.20641118287871850081695350289, −4.98711200728525494019121088558, −4.72737892555716263080807630485, −4.62223264474396551023338912458, −4.27306409071505663171955098285, −4.24636876239150668067996598463, −3.93996871407499258851733672850, −3.48135483660462514839454191964, −3.32518275140735088853897658850, −3.06374886367235219555024990691, −2.89239213837854676139521315483, −2.67891830924081380491204151923, −2.25822468561972766973516447782, −1.94670899284096838851305286727, −1.72164130140141124323637077250, −1.33258430159577161798041547933, −1.23623303611583401379942863921, −0.48708677094200932062148938307, −0.11960259205893057300470421162, 0.11960259205893057300470421162, 0.48708677094200932062148938307, 1.23623303611583401379942863921, 1.33258430159577161798041547933, 1.72164130140141124323637077250, 1.94670899284096838851305286727, 2.25822468561972766973516447782, 2.67891830924081380491204151923, 2.89239213837854676139521315483, 3.06374886367235219555024990691, 3.32518275140735088853897658850, 3.48135483660462514839454191964, 3.93996871407499258851733672850, 4.24636876239150668067996598463, 4.27306409071505663171955098285, 4.62223264474396551023338912458, 4.72737892555716263080807630485, 4.98711200728525494019121088558, 5.20641118287871850081695350289, 5.31321179394805222916478843598, 6.17449691995099878232322613664, 6.22490938668353801448659156404, 6.32074456781066123415077531416, 6.36030411698765390038075524672, 6.88879545586550197711788606706

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