Properties

Label 8-768e4-1.1-c1e4-0-2
Degree $8$
Conductor $347892350976$
Sign $1$
Analytic cond. $1414.33$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·9-s − 20·25-s + 4·49-s − 40·73-s + 27·81-s − 56·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 120·225-s + ⋯
L(s)  = 1  + 2·9-s − 4·25-s + 4/7·49-s − 4.68·73-s + 3·81-s − 5.68·97-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 + 3.38·169-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 + 0.0688·211-s + 0.0669·223-s − 8·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1414.33\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{768} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.630802886\)
\(L(\frac12)\) \(\approx\) \(1.630802886\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
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$C_2^2$ \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.25262851361504783896179659909, −7.13540129664529171036333224101, −7.08100181402466285672416995097, −6.95956252353826936079512097918, −6.52019909053280814191114002591, −6.11256316522981164088914181311, −6.00884931725512877115327692953, −5.82195847638298504143501597175, −5.61409552938894761200238742026, −5.41006058619133677801391830768, −5.01717303883842896772106303660, −4.55296030389065355570238880970, −4.43247110074070987928508827188, −4.40323874543549743438045570322, −3.86836922592150014121603379110, −3.80607652190433463268422928109, −3.74309387477081638887245261975, −3.09756281512473601456889499708, −2.75816835518076104426723789851, −2.62523968266781872014428847440, −1.86883732524368011485336623159, −1.78081088725830406099711628756, −1.67924139145711383148454219131, −1.10152930121939473163336821913, −0.33364763152082249397841607729, 0.33364763152082249397841607729, 1.10152930121939473163336821913, 1.67924139145711383148454219131, 1.78081088725830406099711628756, 1.86883732524368011485336623159, 2.62523968266781872014428847440, 2.75816835518076104426723789851, 3.09756281512473601456889499708, 3.74309387477081638887245261975, 3.80607652190433463268422928109, 3.86836922592150014121603379110, 4.40323874543549743438045570322, 4.43247110074070987928508827188, 4.55296030389065355570238880970, 5.01717303883842896772106303660, 5.41006058619133677801391830768, 5.61409552938894761200238742026, 5.82195847638298504143501597175, 6.00884931725512877115327692953, 6.11256316522981164088914181311, 6.52019909053280814191114002591, 6.95956252353826936079512097918, 7.08100181402466285672416995097, 7.13540129664529171036333224101, 7.25262851361504783896179659909

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