Properties

Label 8-1216e4-1.1-c1e4-0-4
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
Motivic weight $1$
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  + 2·9-s + 12·17-s − 20·25-s + 22·49-s + 32·61-s − 44·73-s − 15·81-s − 72·101-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.91·17-s − 4·25-s + 22/7·49-s + 4.09·61-s − 5.14·73-s − 5/3·81-s − 7.16·101-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.904735809\)
\(L(\frac12)\) \(\approx\) \(1.904735809\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 127 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.05392149383546461651205617317, −6.79531537039988908275275926686, −6.43263947452194859177311187816, −6.38102699494528626698854652711, −5.80847557464670492280227643174, −5.75601524737962496426782283631, −5.63848811164763486835835423818, −5.50658236669242823059194275412, −5.33899556398969600742191701030, −5.18093220608518515839480448259, −4.36818341390879373943825858462, −4.33786618738200936847667330287, −4.22782658630816803377456899954, −4.05305044843527508616730846062, −3.71348254690985208809493685977, −3.45837092799349921579896549348, −3.30717386131226925748647439109, −2.80380046174846143033175828795, −2.56343857070413168442949455301, −2.40196928220026559881875904314, −1.89671423949553437373992045806, −1.57696037327253331610074079133, −1.29127988711736300540013461096, −1.01191301252709051843371456805, −0.29667533820549871869466008955, 0.29667533820549871869466008955, 1.01191301252709051843371456805, 1.29127988711736300540013461096, 1.57696037327253331610074079133, 1.89671423949553437373992045806, 2.40196928220026559881875904314, 2.56343857070413168442949455301, 2.80380046174846143033175828795, 3.30717386131226925748647439109, 3.45837092799349921579896549348, 3.71348254690985208809493685977, 4.05305044843527508616730846062, 4.22782658630816803377456899954, 4.33786618738200936847667330287, 4.36818341390879373943825858462, 5.18093220608518515839480448259, 5.33899556398969600742191701030, 5.50658236669242823059194275412, 5.63848811164763486835835423818, 5.75601524737962496426782283631, 5.80847557464670492280227643174, 6.38102699494528626698854652711, 6.43263947452194859177311187816, 6.79531537039988908275275926686, 7.05392149383546461651205617317

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