Properties

Label 8-1800e4-1.1-c1e4-0-3
Degree $8$
Conductor $1.050\times 10^{13}$
Sign $1$
Analytic cond. $42677.4$
Root an. cond. $3.79118$
Motivic weight $1$
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  − 2·2-s + 2·4-s + 4·7-s − 4·8-s − 8·14-s + 8·16-s − 4·23-s + 8·28-s − 8·31-s − 8·32-s + 8·41-s + 8·46-s + 20·47-s − 12·49-s − 16·56-s + 16·62-s + 8·64-s − 8·71-s − 16·73-s − 32·79-s − 16·82-s − 8·89-s − 8·92-s − 40·94-s + 16·97-s + 24·98-s − 28·103-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.51·7-s − 1.41·8-s − 2.13·14-s + 2·16-s − 0.834·23-s + 1.51·28-s − 1.43·31-s − 1.41·32-s + 1.24·41-s + 1.17·46-s + 2.91·47-s − 1.71·49-s − 2.13·56-s + 2.03·62-s + 64-s − 0.949·71-s − 1.87·73-s − 3.60·79-s − 1.76·82-s − 0.847·89-s − 0.834·92-s − 4.12·94-s + 1.62·97-s + 2.42·98-s − 2.75·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(42677.4\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5173448616\)
\(L(\frac12)\) \(\approx\) \(0.5173448616\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3 \( 1 \)
5 \( 1 \)
good7$D_{4}$ \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$D_{4}$ \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 60 T^{2} + 3446 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.50110993518538510628342355347, −6.49950761551401451318023946466, −6.35342789167998192351429089688, −5.85745124893987466333253314909, −5.77194721436323308792830874430, −5.50413984072621867437478269186, −5.45422510116094619496675468821, −5.33431407809610082571949379537, −4.81287414242273894467219978595, −4.64226078198880089883451875933, −4.45019683532992169697799807571, −4.08047426656114933265909162256, −4.03650263376282594148551657355, −3.69654866653822126223870902057, −3.56501421835816472023405821624, −2.87826714235681184658344459995, −2.84025344071984475779543777462, −2.76062461967093825512111621067, −2.40869432854448879509553710864, −1.97631472798150239608124480479, −1.54681971372107638618552544593, −1.47112066662237462857561876752, −1.34199150637179295573846830704, −0.65892818651162965730428811272, −0.20843093892489246251001890974, 0.20843093892489246251001890974, 0.65892818651162965730428811272, 1.34199150637179295573846830704, 1.47112066662237462857561876752, 1.54681971372107638618552544593, 1.97631472798150239608124480479, 2.40869432854448879509553710864, 2.76062461967093825512111621067, 2.84025344071984475779543777462, 2.87826714235681184658344459995, 3.56501421835816472023405821624, 3.69654866653822126223870902057, 4.03650263376282594148551657355, 4.08047426656114933265909162256, 4.45019683532992169697799807571, 4.64226078198880089883451875933, 4.81287414242273894467219978595, 5.33431407809610082571949379537, 5.45422510116094619496675468821, 5.50413984072621867437478269186, 5.77194721436323308792830874430, 5.85745124893987466333253314909, 6.35342789167998192351429089688, 6.49950761551401451318023946466, 6.50110993518538510628342355347

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