Properties

Label 8-60e8-1.1-c1e4-0-5
Degree $8$
Conductor $1.680\times 10^{14}$
Sign $1$
Analytic cond. $682839.$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·13-s + 32·31-s − 28·37-s + 16·43-s − 48·61-s − 32·67-s + 12·73-s − 20·97-s + 16·103-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 3.32·13-s + 5.74·31-s − 4.60·37-s + 2.43·43-s − 6.14·61-s − 3.90·67-s + 1.40·73-s − 2.03·97-s + 1.57·103-s − 1.81·121-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 + 5.53·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(682839.\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3975835001\)
\(L(\frac12)\) \(\approx\) \(0.3975835001\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 158 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 1666 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 4174 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 5678 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
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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

−6.04668532543660502527654888457, −5.97440971680816426545138835439, −5.56252525130084574154168737374, −5.55400025628583316592498623798, −5.17285351006806805083530414906, −4.89273061690963016493473311421, −4.76641505781645766909473015919, −4.56740093947293150311505605438, −4.52014291113320582051759004203, −4.40064059063919626550316417830, −4.34712062874239944329647649561, −3.67943489776109016588564801614, −3.55662933307974575407777360332, −3.17360639237138299437903135496, −2.93211582289602428292844811327, −2.90866945676150514639396811472, −2.82254397250739919130189069978, −2.37740877097733634816362643243, −2.29089405216119942479353105434, −1.80322266465904047654941160944, −1.71838982383836862301348600696, −1.30288202343338596925516312867, −1.04013909879030420156710336846, −0.51720286077981030061341818356, −0.12056875026187638384639040697, 0.12056875026187638384639040697, 0.51720286077981030061341818356, 1.04013909879030420156710336846, 1.30288202343338596925516312867, 1.71838982383836862301348600696, 1.80322266465904047654941160944, 2.29089405216119942479353105434, 2.37740877097733634816362643243, 2.82254397250739919130189069978, 2.90866945676150514639396811472, 2.93211582289602428292844811327, 3.17360639237138299437903135496, 3.55662933307974575407777360332, 3.67943489776109016588564801614, 4.34712062874239944329647649561, 4.40064059063919626550316417830, 4.52014291113320582051759004203, 4.56740093947293150311505605438, 4.76641505781645766909473015919, 4.89273061690963016493473311421, 5.17285351006806805083530414906, 5.55400025628583316592498623798, 5.56252525130084574154168737374, 5.97440971680816426545138835439, 6.04668532543660502527654888457

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