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Properties

Label 8-3584e4-1.1-c0e4-0-0
Degree $8$
Conductor $1.650\times 10^{14}$
Sign $1$
Analytic cond. $10.2352$
Root an. cond. $1.33740$
Motivic weight $0$
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  − 4·11-s − 4·23-s − 4·29-s − 4·37-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯
L(s)  = 1  − 4·11-s − 4·23-s − 4·29-s − 4·37-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(10.2352\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1312418719\)
\(L(\frac12)\) \(\approx\) \(0.1312418719\)
\(L(1)\) 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 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.06486796737626047108309702127, −5.92006208450241034188729142527, −5.65278996164333649935611356501, −5.54462514033664121018387147239, −5.49851462577772948782722030414, −5.41928835519669893166588448630, −5.32818343898404828859655883619, −4.74231438930181858769852917685, −4.56488312573743181669197896855, −4.50302636536104243442070786687, −4.43590779421409828352644563031, −3.80511738776761967665269351208, −3.66918454437157737967395871898, −3.63835955895356237986624984866, −3.41710713307571425326477592164, −3.07791684451694665618487169571, −2.97399246043269776412029238366, −2.44864436639416112236042640109, −2.36918018122863424262731986994, −2.03792795615319591688982556547, −1.91762312295104800324687548662, −1.70751372482786433314108241827, −1.68856494803494120002271164551, −0.57781074874493026118869097139, −0.18489734317147735298498070942, 0.18489734317147735298498070942, 0.57781074874493026118869097139, 1.68856494803494120002271164551, 1.70751372482786433314108241827, 1.91762312295104800324687548662, 2.03792795615319591688982556547, 2.36918018122863424262731986994, 2.44864436639416112236042640109, 2.97399246043269776412029238366, 3.07791684451694665618487169571, 3.41710713307571425326477592164, 3.63835955895356237986624984866, 3.66918454437157737967395871898, 3.80511738776761967665269351208, 4.43590779421409828352644563031, 4.50302636536104243442070786687, 4.56488312573743181669197896855, 4.74231438930181858769852917685, 5.32818343898404828859655883619, 5.41928835519669893166588448630, 5.49851462577772948782722030414, 5.54462514033664121018387147239, 5.65278996164333649935611356501, 5.92006208450241034188729142527, 6.06486796737626047108309702127

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