Properties

Label 8-1232e4-1.1-c1e4-0-3
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $9365.93$
Root an. cond. $3.13649$
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 + 12·11-s + 12·23-s + 10·25-s − 4·37-s − 4·49-s − 44·67-s − 36·71-s − 15·81-s + 24·99-s − 12·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s + 3.61·11-s + 2.50·23-s + 2·25-s − 0.657·37-s − 4/7·49-s − 5.37·67-s − 4.27·71-s − 5/3·81-s + 2.41·99-s − 1.12·113-s + 7.81·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(9365.93\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.978599477\)
\(L(\frac12)\) \(\approx\) \(4.978599477\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 173 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 149 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.97792650558364186517294018599, −6.82531273143171045690050860177, −6.55066269547999907306273893163, −6.28617455634697364319278079491, −6.10756877235843440314977399119, −6.01575067939305668588878806518, −5.66347128252810403950060603700, −5.42196545087892792511555714865, −4.95873334135115377514803632250, −4.92412785900335810059070846444, −4.51331879040944607046178919494, −4.50440851723466944615809731540, −4.18545337933987888266949729514, −4.07996347798920515985039909355, −3.75256782382810686277329615839, −3.19537056146018256552951646254, −3.18366011834878477684170431743, −3.11677516092445479752027791405, −2.70516626812986673353278327861, −2.34658427216139065751626210794, −1.56341748315225492556090834085, −1.43929644792923261774976075637, −1.32338818904276140989184174216, −1.27982278696716040161900821061, −0.45582375148949135865595755256, 0.45582375148949135865595755256, 1.27982278696716040161900821061, 1.32338818904276140989184174216, 1.43929644792923261774976075637, 1.56341748315225492556090834085, 2.34658427216139065751626210794, 2.70516626812986673353278327861, 3.11677516092445479752027791405, 3.18366011834878477684170431743, 3.19537056146018256552951646254, 3.75256782382810686277329615839, 4.07996347798920515985039909355, 4.18545337933987888266949729514, 4.50440851723466944615809731540, 4.51331879040944607046178919494, 4.92412785900335810059070846444, 4.95873334135115377514803632250, 5.42196545087892792511555714865, 5.66347128252810403950060603700, 6.01575067939305668588878806518, 6.10756877235843440314977399119, 6.28617455634697364319278079491, 6.55066269547999907306273893163, 6.82531273143171045690050860177, 6.97792650558364186517294018599

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