Properties

Label 8-3072e4-1.1-c0e4-0-4
Degree $8$
Conductor $8.906\times 10^{13}$
Sign $1$
Analytic cond. $5.52475$
Root an. cond. $1.23819$
Motivic weight $0$
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  + 4·5-s + 8·25-s + 4·29-s + 4·53-s − 81-s − 8·97-s + 4·101-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4·5-s + 8·25-s + 4·29-s + 4·53-s − 81-s − 8·97-s + 4·101-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5.52475\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3072} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} \cdot 3^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.151445737\)
\(L(\frac12)\) \(\approx\) \(5.151445737\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
good5$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_1$ \( ( 1 + T )^{8} \)
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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

−6.30316039865065811692633334538, −6.17259652628470540877831807922, −5.91670545090749496973089760558, −5.83972136251799428852648814744, −5.53244551160912387720814522941, −5.48624913155813880209990266209, −5.16052111905534477347100105754, −5.03078800469277393504516950514, −4.96602860971618869624204441395, −4.61715759243337261069415793406, −4.30521829959841082469608134852, −4.13468988420988663562583351683, −4.05031999766707573593851654903, −3.46369972840480522571364315768, −3.42409462183955206607210737542, −2.86688605358286930455247893522, −2.81839659550972779658887060070, −2.54116998413610041546289081331, −2.49897956678911766965327139983, −2.17995600944609962171415975583, −2.05239348523040099847752966227, −1.63983652320964928946055910882, −1.18769800524870273713950342010, −1.15830482563995193885761785435, −1.01078294170993759154394353789, 1.01078294170993759154394353789, 1.15830482563995193885761785435, 1.18769800524870273713950342010, 1.63983652320964928946055910882, 2.05239348523040099847752966227, 2.17995600944609962171415975583, 2.49897956678911766965327139983, 2.54116998413610041546289081331, 2.81839659550972779658887060070, 2.86688605358286930455247893522, 3.42409462183955206607210737542, 3.46369972840480522571364315768, 4.05031999766707573593851654903, 4.13468988420988663562583351683, 4.30521829959841082469608134852, 4.61715759243337261069415793406, 4.96602860971618869624204441395, 5.03078800469277393504516950514, 5.16052111905534477347100105754, 5.48624913155813880209990266209, 5.53244551160912387720814522941, 5.83972136251799428852648814744, 5.91670545090749496973089760558, 6.17259652628470540877831807922, 6.30316039865065811692633334538

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