Properties

Label 8-48e8-1.1-c2e4-0-13
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  − 80·19-s − 8·25-s + 160·43-s + 20·49-s − 400·67-s − 80·73-s + 160·97-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 460·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4.21·19-s − 0.319·25-s + 3.72·43-s + 0.408·49-s − 5.97·67-s − 1.09·73-s + 1.64·97-s + 2.80·121-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 − 2.72·169-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 + 0.00448·223-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

\(L(\frac{3}{2})\) \(\approx\) \(1.016831232\)
\(L(\frac12)\) \(\approx\) \(1.016831232\)
\(L(2)\) 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$C_2^2$ \( ( 1 + 4 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 230 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 128 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 332 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 778 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 1840 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 982 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 4268 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 5810 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
67$C_2$ \( ( 1 + 100 T + p^{2} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 4682 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 9782 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 2422 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{4} \)
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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

−6.06031069412633675808882902932, −6.05946304455884958806522186364, −6.05039601933368998342131175456, −5.93155442862215245174799442768, −5.17322695514098306319998599112, −5.16293701910768687090895482061, −5.06096050539795166328913678520, −4.52036191493287287537996850941, −4.37936103391565022821156179382, −4.23580639065801223972631129085, −4.19395666683100672097590801666, −4.13727471419766797354972718425, −3.55404649740844485069620985038, −3.45674327454576437847480472955, −3.07243265542913663496465927939, −2.80799604569383465595993447916, −2.54211969231401446098746950977, −2.36341095923150852238131432571, −2.21885677719001832908850254206, −1.74839264346501482224174085955, −1.66382956466558677427684278933, −1.28980503429743713066219381100, −0.874446351895503344352321540752, −0.41339658558687446656873549677, −0.17406383151243383918410930347, 0.17406383151243383918410930347, 0.41339658558687446656873549677, 0.874446351895503344352321540752, 1.28980503429743713066219381100, 1.66382956466558677427684278933, 1.74839264346501482224174085955, 2.21885677719001832908850254206, 2.36341095923150852238131432571, 2.54211969231401446098746950977, 2.80799604569383465595993447916, 3.07243265542913663496465927939, 3.45674327454576437847480472955, 3.55404649740844485069620985038, 4.13727471419766797354972718425, 4.19395666683100672097590801666, 4.23580639065801223972631129085, 4.37936103391565022821156179382, 4.52036191493287287537996850941, 5.06096050539795166328913678520, 5.16293701910768687090895482061, 5.17322695514098306319998599112, 5.93155442862215245174799442768, 6.05039601933368998342131175456, 6.05946304455884958806522186364, 6.06031069412633675808882902932

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