Properties

Label 8-825e4-1.1-c1e4-0-2
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
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  − 8·4-s + 5·9-s + 40·16-s − 20·31-s − 40·36-s + 28·49-s − 160·64-s + 16·81-s − 22·121-s + 160·124-s + 127-s + 131-s + 137-s + 139-s + 200·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 224·196-s + ⋯
L(s)  = 1  − 4·4-s + 5/3·9-s + 10·16-s − 3.59·31-s − 6.66·36-s + 4·49-s − 20·64-s + 16/9·81-s − 2·121-s + 14.3·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 50/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 16·196-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6447436680\)
\(L(\frac12)\) \(\approx\) \(0.6447436680\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
5 \( 1 \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( ( 1 + 95 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.43181015916288493113809601077, −7.27880225648591179336223759098, −6.97113932821447455354628354830, −6.66616626958352135965378349550, −6.50782503277666358392566002653, −5.85178239071061506736739649709, −5.69128670001264019738298771088, −5.61078433429322471167381449637, −5.39624003169409678417574263387, −5.35031418448083963065659489179, −4.76763387295287100738563281793, −4.75782028154387791004990478000, −4.41297941920619156399297024038, −4.18076235095347934769139061450, −4.07750982349987772190396743637, −3.80737018210525945412835870765, −3.61731115066062440854324804012, −3.39934128202570735807086305435, −3.05999526166417652689461231315, −2.50011227736513341061658032683, −1.96420001903979055192800385570, −1.60823952968475727898726893515, −1.18082830095439656368622583428, −0.792718094811562857389913547233, −0.33467337291877393707617747332, 0.33467337291877393707617747332, 0.792718094811562857389913547233, 1.18082830095439656368622583428, 1.60823952968475727898726893515, 1.96420001903979055192800385570, 2.50011227736513341061658032683, 3.05999526166417652689461231315, 3.39934128202570735807086305435, 3.61731115066062440854324804012, 3.80737018210525945412835870765, 4.07750982349987772190396743637, 4.18076235095347934769139061450, 4.41297941920619156399297024038, 4.75782028154387791004990478000, 4.76763387295287100738563281793, 5.35031418448083963065659489179, 5.39624003169409678417574263387, 5.61078433429322471167381449637, 5.69128670001264019738298771088, 5.85178239071061506736739649709, 6.50782503277666358392566002653, 6.66616626958352135965378349550, 6.97113932821447455354628354830, 7.27880225648591179336223759098, 7.43181015916288493113809601077

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