Properties

Label 8-1216e4-1.1-c1e4-0-7
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  + 4·9-s + 20·17-s − 2·25-s − 24·41-s − 6·49-s − 36·73-s − 6·81-s − 16·89-s − 48·97-s + 72·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 80·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4/3·9-s + 4.85·17-s − 2/5·25-s − 3.74·41-s − 6/7·49-s − 4.21·73-s − 2/3·81-s − 1.69·89-s − 4.87·97-s + 6.77·113-s − 0.545·121-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 + 6.46·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

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

\(L(1)\) \(\approx\) \(3.208234946\)
\(L(\frac12)\) \(\approx\) \(3.208234946\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \)
7$C_2^2$ \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 23 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 9 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
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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

−7.03597694554568477782925793613, −6.76524858234280497576853962935, −6.67451998530025654023698139397, −6.29917760001774489458982200150, −5.89138904798455870273502042289, −5.73709645719161143310994940025, −5.61312760757611456318205966009, −5.52161813586427681127493154043, −5.30005584770578324381022678625, −4.82318302801540567901277684336, −4.80729650171502837973667391370, −4.42154850135880767583686358960, −4.12950482803184405001146979328, −4.03421058638350864897326894513, −3.65612567421964293048161167803, −3.19805394882484495721313956110, −3.17598049585244861117592391403, −3.01287100473840157779367045948, −3.00870965464448832371541584128, −2.08225422170575501897262664898, −1.86826817566729217909127306176, −1.50571303898887487008178547615, −1.26258727654046097602948520640, −1.17055033005542009634975040100, −0.37043274595792950317024668851, 0.37043274595792950317024668851, 1.17055033005542009634975040100, 1.26258727654046097602948520640, 1.50571303898887487008178547615, 1.86826817566729217909127306176, 2.08225422170575501897262664898, 3.00870965464448832371541584128, 3.01287100473840157779367045948, 3.17598049585244861117592391403, 3.19805394882484495721313956110, 3.65612567421964293048161167803, 4.03421058638350864897326894513, 4.12950482803184405001146979328, 4.42154850135880767583686358960, 4.80729650171502837973667391370, 4.82318302801540567901277684336, 5.30005584770578324381022678625, 5.52161813586427681127493154043, 5.61312760757611456318205966009, 5.73709645719161143310994940025, 5.89138904798455870273502042289, 6.29917760001774489458982200150, 6.67451998530025654023698139397, 6.76524858234280497576853962935, 7.03597694554568477782925793613

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