Properties

Label 8-2e36-1.1-c2e4-0-1
Degree $8$
Conductor $68719476736$
Sign $1$
Analytic cond. $37880.8$
Root an. cond. $3.73510$
Motivic weight $2$
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  − 100·25-s − 184·41-s + 196·49-s − 34·81-s − 392·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 4·25-s − 4.48·41-s + 4·49-s − 0.419·81-s − 3.46·113-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 − 4·169-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 + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36}\)
Sign: $1$
Analytic conductor: \(37880.8\)
Root analytic conductor: \(3.73510\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1618257165\)
\(L(\frac12)\) \(\approx\) \(0.1618257165\)
\(L(2)\) 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 \( 1 \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} ) \)
5$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
11$C_2^2$$\times$$C_2^2$ \( ( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} ) \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 574 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
37$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
41$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{4} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - 120 T + 7200 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )( 1 + 120 T + 7200 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
59$C_2^2$$\times$$C_2^2$ \( ( 1 - 120 T + 7200 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )( 1 + 120 T + 7200 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} ) \)
61$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 168 T + 14112 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} )( 1 + 168 T + 14112 T^{2} + 168 p^{2} T^{3} + p^{4} T^{4} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2^2$ \( ( 1 + 9506 T^{2} + p^{4} T^{4} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 72 T + 2592 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )( 1 + 72 T + 2592 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} ) \)
89$C_2^2$ \( ( 1 + 5474 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 9982 T^{2} + p^{4} T^{4} )^{2} \)
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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

−7.54496201369519345856615396567, −7.49362089085268365121486626597, −7.06159412350939683916441950433, −7.02284354612260863186035728462, −6.75285754624906009430192148529, −6.48048178937702194810551889683, −6.05266609481204406546807013447, −5.90663477350550173092647195708, −5.61085326408028744582661215445, −5.53204424800892812733793805725, −5.29059240128494167249971274028, −4.84614562490900313773014768993, −4.61039881817635448747192758121, −4.20976214986503029283208650376, −4.04567183131378433833147501824, −3.73589701087092884285629259680, −3.47291560065710072051790045442, −3.30970439351563337479392992959, −2.74736990690413497296960595414, −2.34024367121887577327266669878, −2.11033319827092539985263490734, −1.74850958804639383578086825162, −1.45923902308736539313932287318, −0.78092502109633740070459906789, −0.083407147722585489702499989776, 0.083407147722585489702499989776, 0.78092502109633740070459906789, 1.45923902308736539313932287318, 1.74850958804639383578086825162, 2.11033319827092539985263490734, 2.34024367121887577327266669878, 2.74736990690413497296960595414, 3.30970439351563337479392992959, 3.47291560065710072051790045442, 3.73589701087092884285629259680, 4.04567183131378433833147501824, 4.20976214986503029283208650376, 4.61039881817635448747192758121, 4.84614562490900313773014768993, 5.29059240128494167249971274028, 5.53204424800892812733793805725, 5.61085326408028744582661215445, 5.90663477350550173092647195708, 6.05266609481204406546807013447, 6.48048178937702194810551889683, 6.75285754624906009430192148529, 7.02284354612260863186035728462, 7.06159412350939683916441950433, 7.49362089085268365121486626597, 7.54496201369519345856615396567

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