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Properties

Label 8-8470e4-1.1-c1e4-0-4
Degree $8$
Conductor $5.147\times 10^{15}$
Sign $1$
Analytic cond. $2.09238\times 10^{7}$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 4·3-s + 10·4-s + 4·5-s − 16·6-s − 4·7-s + 20·8-s + 2·9-s + 16·10-s − 40·12-s − 16·14-s − 16·15-s + 35·16-s + 8·17-s + 8·18-s − 12·19-s + 40·20-s + 16·21-s − 8·23-s − 80·24-s + 10·25-s + 16·27-s − 40·28-s − 8·29-s − 64·30-s + 56·32-s + 32·34-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 1.51·7-s + 7.07·8-s + 2/3·9-s + 5.05·10-s − 11.5·12-s − 4.27·14-s − 4.13·15-s + 35/4·16-s + 1.94·17-s + 1.88·18-s − 2.75·19-s + 8.94·20-s + 3.49·21-s − 1.66·23-s − 16.3·24-s + 2·25-s + 3.07·27-s − 7.55·28-s − 1.48·29-s − 11.6·30-s + 9.89·32-s + 5.48·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.09238\times 10^{7}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good3$D_{4}$ \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 30 T^{2} + 48 T^{3} + 419 T^{4} + 48 p T^{5} + 30 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 12 T + 102 T^{2} + 624 T^{3} + 2987 T^{4} + 624 p T^{5} + 102 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 82 T^{2} + 544 T^{3} + 2715 T^{4} + 544 p T^{5} + 82 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 100 T^{2} + 664 T^{3} + 4134 T^{4} + 664 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^3$ \( 1 + 194 T^{4} + p^{4} T^{8} \)
37$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 16 T + 204 T^{2} + 1616 T^{3} + 11558 T^{4} + 1616 p T^{5} + 204 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
41$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 8 T + 80 T^{2} - 8 T^{3} + 802 T^{4} - 8 p T^{5} + 80 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
43$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 112 T^{2} + 728 T^{3} + 5650 T^{4} + 728 p T^{5} + 112 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 16 T + 184 T^{2} + 1712 T^{3} + 13650 T^{4} + 1712 p T^{5} + 184 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
53$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 136 T^{2} - 192 T^{3} + 8802 T^{4} - 192 p T^{5} + 136 p^{2} T^{6} + p^{4} T^{8} \)
59$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 218 T^{2} + 1280 T^{3} + 18835 T^{4} + 1280 p T^{5} + 218 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
61$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 228 T^{2} + 1336 T^{3} + 20246 T^{4} + 1336 p T^{5} + 228 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 136 T^{2} + 10146 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} \)
71$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 92 T^{2} - 280 T^{3} - 26 T^{4} - 280 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 4 T + 148 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 24 T + 6 p T^{2} + 5760 T^{3} + 61379 T^{4} + 5760 p T^{5} + 6 p^{3} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
83$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 8 T + 50 T^{2} + 640 T^{3} - 11285 T^{4} + 640 p T^{5} + 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 208 T^{2} + 21858 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} \)
97$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 244 T^{2} + 824 T^{3} + 25846 T^{4} + 824 p T^{5} + 244 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.96524976082356566283673664467, −5.51661880797613135613941570897, −5.47579644128930334928571994644, −5.33151121290417656266965437426, −5.26454576410301985455177157101, −5.04185239532623729312200667249, −4.91623703981130404514698527926, −4.67466461003413361608446759902, −4.47719188375516881382919912542, −4.16623705037890174437685079831, −3.99895721646085089034625954409, −3.80430401437602265860659752393, −3.67994147396936548168109004166, −3.43145572254642024806820909105, −3.25550862994466498584217813827, −3.04224701828305115864332014731, −2.92564037308280975596712915890, −2.52495436365313852298061959061, −2.34616646873918088868676710116, −2.32549793410702565266513341996, −2.11178366081477015658998927270, −1.56624819212278216523012407055, −1.51433848791891405573202105470, −1.26307135938770776399751758257, −1.15925529896471374533291926322, 0, 0, 0, 0, 1.15925529896471374533291926322, 1.26307135938770776399751758257, 1.51433848791891405573202105470, 1.56624819212278216523012407055, 2.11178366081477015658998927270, 2.32549793410702565266513341996, 2.34616646873918088868676710116, 2.52495436365313852298061959061, 2.92564037308280975596712915890, 3.04224701828305115864332014731, 3.25550862994466498584217813827, 3.43145572254642024806820909105, 3.67994147396936548168109004166, 3.80430401437602265860659752393, 3.99895721646085089034625954409, 4.16623705037890174437685079831, 4.47719188375516881382919912542, 4.67466461003413361608446759902, 4.91623703981130404514698527926, 5.04185239532623729312200667249, 5.26454576410301985455177157101, 5.33151121290417656266965437426, 5.47579644128930334928571994644, 5.51661880797613135613941570897, 5.96524976082356566283673664467

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