Properties

Label 8-300e4-1.1-c2e4-0-8
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 136·19-s + 56·31-s + 188·49-s − 184·61-s + 88·79-s − 77·81-s − 344·109-s + 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-s + 272·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2/9·9-s + 7.15·19-s + 1.80·31-s + 3.83·49-s − 3.01·61-s + 1.11·79-s − 0.950·81-s − 3.15·109-s + 1.02·121-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 + 3.24·169-s + 1.59·171-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

\(L(\frac{3}{2})\) \(\approx\) \(6.042778892\)
\(L(\frac12)\) \(\approx\) \(6.042778892\)
\(L(2)\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 398 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 562 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 398 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2642 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 3634 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 2798 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 3998 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6782 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 7954 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 578 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( ( 1 - 3934 T^{2} + p^{4} T^{4} )^{2} \)
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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

−8.141673580499715912496144123306, −7.84782260458817317202938281570, −7.81485201965005719502607699608, −7.47909025193132483199793278401, −7.38465657961656225191887835176, −7.17529506865639819115084179635, −6.88568984473372541655652726549, −6.38054458799849057792642677908, −6.37601401955329073165337762089, −5.51784158888108776012118989263, −5.50584447824850053448546274877, −5.49875699226628949783339057997, −5.42653055467533229033956953344, −4.76656131635516598791009424454, −4.41930504001647050265224355908, −4.33337270366909505230202698452, −3.67595229995746786862073002174, −3.38617063538297609747078151090, −3.13385976447391963677200149841, −2.83405182838099226883775213756, −2.72152468322534533510698944487, −1.87663528629176733379438238761, −1.25078637838298401631685047854, −0.996311634514038622026371343584, −0.77475133942346317145728847786, 0.77475133942346317145728847786, 0.996311634514038622026371343584, 1.25078637838298401631685047854, 1.87663528629176733379438238761, 2.72152468322534533510698944487, 2.83405182838099226883775213756, 3.13385976447391963677200149841, 3.38617063538297609747078151090, 3.67595229995746786862073002174, 4.33337270366909505230202698452, 4.41930504001647050265224355908, 4.76656131635516598791009424454, 5.42653055467533229033956953344, 5.49875699226628949783339057997, 5.50584447824850053448546274877, 5.51784158888108776012118989263, 6.37601401955329073165337762089, 6.38054458799849057792642677908, 6.88568984473372541655652726549, 7.17529506865639819115084179635, 7.38465657961656225191887835176, 7.47909025193132483199793278401, 7.81485201965005719502607699608, 7.84782260458817317202938281570, 8.141673580499715912496144123306

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