Properties

Label 8-930e4-1.1-c1e4-0-5
Degree $8$
Conductor $748052010000$
Sign $1$
Analytic cond. $3041.16$
Root an. cond. $2.72508$
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  − 2·4-s − 16·7-s + 3·16-s − 8·19-s − 2·25-s + 32·28-s + 20·31-s + 132·49-s − 4·64-s + 16·67-s + 16·76-s − 9·81-s + 56·97-s + 4·100-s + 16·103-s − 64·109-s − 48·112-s + 4·121-s − 40·124-s + 127-s + 131-s + 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 4-s − 6.04·7-s + 3/4·16-s − 1.83·19-s − 2/5·25-s + 6.04·28-s + 3.59·31-s + 18.8·49-s − 1/2·64-s + 1.95·67-s + 1.83·76-s − 81-s + 5.68·97-s + 2/5·100-s + 1.57·103-s − 6.13·109-s − 4.53·112-s + 4/11·121-s − 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 11.0·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(3041.16\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4893617612\)
\(L(\frac12)\) \(\approx\) \(0.4893617612\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
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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

−7.06504132842321946108501167189, −6.65824643592717563787413846397, −6.63484674152703722280720964158, −6.49812157135989280931032658214, −6.47700255850120500820894218825, −6.28403738718995254044774298390, −5.93736066453395959914922735478, −5.74268044766318991151298087838, −5.58541357955660783925553677735, −5.19668567932328265085813598065, −4.85181410486138855101146471851, −4.45656208179113712495322328784, −4.25184918460406237006618727338, −3.95417809015236702877811425878, −3.95061852121690728800583033493, −3.56935227814330193844270139000, −3.13479123036722053212503588555, −3.01699969319885962967433074306, −3.01230036477294194359640859200, −2.75901749428737794613219065637, −2.16808448627865159263010589620, −2.01208552782541318451489267142, −0.892895951309314894610010250955, −0.49408779239236033858723886682, −0.41624285938825664574505994309, 0.41624285938825664574505994309, 0.49408779239236033858723886682, 0.892895951309314894610010250955, 2.01208552782541318451489267142, 2.16808448627865159263010589620, 2.75901749428737794613219065637, 3.01230036477294194359640859200, 3.01699969319885962967433074306, 3.13479123036722053212503588555, 3.56935227814330193844270139000, 3.95061852121690728800583033493, 3.95417809015236702877811425878, 4.25184918460406237006618727338, 4.45656208179113712495322328784, 4.85181410486138855101146471851, 5.19668567932328265085813598065, 5.58541357955660783925553677735, 5.74268044766318991151298087838, 5.93736066453395959914922735478, 6.28403738718995254044774298390, 6.47700255850120500820894218825, 6.49812157135989280931032658214, 6.63484674152703722280720964158, 6.65824643592717563787413846397, 7.06504132842321946108501167189

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