Properties

Label 8-300e4-1.1-c2e4-0-1
Degree $8$
Conductor $8100000000$
Sign $1$
Analytic cond. $4465.03$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 17·9-s − 44·19-s − 184·31-s + 68·49-s − 64·61-s − 272·79-s + 208·81-s − 704·109-s − 146·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 668·169-s − 748·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 17/9·9-s − 2.31·19-s − 5.93·31-s + 1.38·49-s − 1.04·61-s − 3.44·79-s + 2.56·81-s − 6.45·109-s − 1.20·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.95·169-s − 4.37·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(4465.03\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1391367915\)
\(L(\frac12)\) \(\approx\) \(0.1391367915\)
\(L(2)\) 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$C_2^2$ \( 1 - 17 T^{2} + p^{4} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 13 T + p^{2} T^{2} )^{2}( 1 + 13 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 263 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 11 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 202 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 422 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 527 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 3158 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 4358 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 1922 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 3791 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 1258 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 457 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 68 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 13463 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 13007 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 18334 T^{2} + p^{4} T^{4} )^{2} \)
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

−8.372934043276650756535241827613, −7.84356027685349237923237774835, −7.77840162007580419900345731076, −7.66500390149678367963005145053, −6.99626340442483195494221904805, −6.98575358375508349243388827645, −6.95942774827552603533961551296, −6.77152207805068020533276548506, −6.14906916111516207060160817382, −5.90526927732681388990142879591, −5.58088591713682884143870267690, −5.39008185217622609256893890710, −5.26040036256378964311798677800, −4.60609002384745704697562082634, −4.33569133529852055960664991271, −4.14065181655108974557940821923, −3.85639682646753472871634804551, −3.82282415323021516604790355313, −3.09538553677403797475582856573, −2.89701602767129349658174197167, −2.09128915811338465236388886178, −1.83328191531635504418544631323, −1.80903108124915669021394032373, −1.15344968064700163055068012612, −0.085071757099238205817910951762, 0.085071757099238205817910951762, 1.15344968064700163055068012612, 1.80903108124915669021394032373, 1.83328191531635504418544631323, 2.09128915811338465236388886178, 2.89701602767129349658174197167, 3.09538553677403797475582856573, 3.82282415323021516604790355313, 3.85639682646753472871634804551, 4.14065181655108974557940821923, 4.33569133529852055960664991271, 4.60609002384745704697562082634, 5.26040036256378964311798677800, 5.39008185217622609256893890710, 5.58088591713682884143870267690, 5.90526927732681388990142879591, 6.14906916111516207060160817382, 6.77152207805068020533276548506, 6.95942774827552603533961551296, 6.98575358375508349243388827645, 6.99626340442483195494221904805, 7.66500390149678367963005145053, 7.77840162007580419900345731076, 7.84356027685349237923237774835, 8.372934043276650756535241827613

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