Properties

Label 8-48e8-1.1-c2e4-0-28
Degree $8$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $1.55335\times 10^{7}$
Root an. cond. $7.92334$
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  − 40·17-s + 92·25-s + 232·41-s + 100·49-s + 184·73-s + 328·89-s + 8·97-s + 440·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 668·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.35·17-s + 3.67·25-s + 5.65·41-s + 2.04·49-s + 2.52·73-s + 3.68·89-s + 8/97·97-s + 3.89·113-s − 3.20·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.95·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.55335\times 10^{7}\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(10.67343352\)
\(L(\frac12)\) \(\approx\) \(10.67343352\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1006 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1874 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 58 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 1346 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 382 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 1150 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 6766 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 8930 T^{2} + p^{4} T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2$ \( ( 1 - 46 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 1390 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 11426 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{4} \)
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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

−6.44345903248389114371277421086, −5.91838537362366870714874660656, −5.90395211089713032119940164207, −5.54890409103611305397396946473, −5.44679344780257172217878757204, −4.97424748181450191290868539251, −4.88075447761143464262489334074, −4.80278860746713486783798607421, −4.62990701588148983351473568654, −4.11345153784203473493615048325, −4.08486040669370148244136438803, −3.92708603298862517069102243335, −3.91480306672593296944873825757, −3.13586598963087078092760774355, −3.01108482372970073198034129232, −2.94807765796001552132122540498, −2.66379287857066629605144509506, −2.29457791482618979114251267847, −2.15957543020744756516764245928, −2.01791558252308334864128376231, −1.57628357372537341357225596994, −0.894291396568420371438610688122, −0.823185506648304492654521347824, −0.73248850573389047262912518206, −0.45895185726192935375148464568, 0.45895185726192935375148464568, 0.73248850573389047262912518206, 0.823185506648304492654521347824, 0.894291396568420371438610688122, 1.57628357372537341357225596994, 2.01791558252308334864128376231, 2.15957543020744756516764245928, 2.29457791482618979114251267847, 2.66379287857066629605144509506, 2.94807765796001552132122540498, 3.01108482372970073198034129232, 3.13586598963087078092760774355, 3.91480306672593296944873825757, 3.92708603298862517069102243335, 4.08486040669370148244136438803, 4.11345153784203473493615048325, 4.62990701588148983351473568654, 4.80278860746713486783798607421, 4.88075447761143464262489334074, 4.97424748181450191290868539251, 5.44679344780257172217878757204, 5.54890409103611305397396946473, 5.90395211089713032119940164207, 5.91838537362366870714874660656, 6.44345903248389114371277421086

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