Properties

Label 8-96e8-1.1-c1e4-0-17
Degree $8$
Conductor $7.214\times 10^{15}$
Sign $1$
Analytic cond. $2.93277\times 10^{7}$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·25-s − 64·67-s − 44·121-s + 127-s + 131-s + 137-s + 139-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 + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 4·25-s − 7.81·67-s − 4·121-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 + 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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.93277\times 10^{7}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{4} \)
7$C_2^3$ \( 1 - 94 T^{4} + p^{4} T^{8} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2^3$ \( 1 + 146 T^{4} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^3$ \( 1 + 194 T^{4} + p^{4} T^{8} \)
37$C_2^3$ \( 1 - 2062 T^{4} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^3$ \( 1 - 1966 T^{4} + p^{4} T^{8} \)
67$C_2$ \( ( 1 + 16 T + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^3$ \( 1 + 7682 T^{4} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.80364309809751450505556660585, −5.75083672563839655216111551405, −5.33346530256710295936851411287, −5.26227966166048988244772824239, −5.21114722170712189196749862302, −4.69973857411545586297705161694, −4.65148409012182754932447750642, −4.50558026123186279143574713940, −4.25160323299228911034355326772, −4.21742800480488082328991502184, −3.85631421440277869096500323868, −3.81538375692025753991748620165, −3.63510243770382464811985115218, −3.29487586134192794883138674384, −3.11958349431956666006985334246, −2.99891639345788159405111319893, −2.83198476199136641689326549927, −2.34404997446115660914470319164, −2.27196005744173555007763847150, −2.09787078974469372903157306254, −2.02828459907767621748096648731, −1.43425819682647374049095168760, −1.34925923964010256885618909609, −1.21377998738872618041691397987, −1.08862081458042372228834389055, 0, 0, 0, 0, 1.08862081458042372228834389055, 1.21377998738872618041691397987, 1.34925923964010256885618909609, 1.43425819682647374049095168760, 2.02828459907767621748096648731, 2.09787078974469372903157306254, 2.27196005744173555007763847150, 2.34404997446115660914470319164, 2.83198476199136641689326549927, 2.99891639345788159405111319893, 3.11958349431956666006985334246, 3.29487586134192794883138674384, 3.63510243770382464811985115218, 3.81538375692025753991748620165, 3.85631421440277869096500323868, 4.21742800480488082328991502184, 4.25160323299228911034355326772, 4.50558026123186279143574713940, 4.65148409012182754932447750642, 4.69973857411545586297705161694, 5.21114722170712189196749862302, 5.26227966166048988244772824239, 5.33346530256710295936851411287, 5.75083672563839655216111551405, 5.80364309809751450505556660585

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