Properties

Label 8-2475e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.752\times 10^{13}$
Sign $1$
Analytic cond. $152548.$
Root an. cond. $4.44555$
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  + 2·4-s + 4·11-s − 5·16-s − 8·19-s − 16·31-s + 8·44-s + 20·49-s + 8·61-s − 20·64-s − 16·76-s + 40·79-s − 24·89-s + 40·109-s + 10·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s − 20·176-s + ⋯
L(s)  = 1  + 4-s + 1.20·11-s − 5/4·16-s − 1.83·19-s − 2.87·31-s + 1.20·44-s + 20/7·49-s + 1.02·61-s − 5/2·64-s − 1.83·76-s + 4.50·79-s − 2.54·89-s + 3.83·109-s + 0.909·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s − 1.50·176-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(152548.\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.671988462\)
\(L(\frac12)\) \(\approx\) \(1.671988462\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{4} \)
good2$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$D_{4}$ \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 68 T^{2} + 4086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 44 T^{2} - 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 68 T^{2} + 10086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 20 T^{2} + 6966 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.32630323042012411979064576657, −6.21119806104241908709911456345, −6.15477963843348821495211188053, −5.69490244023560750193203742191, −5.49159182937081071445558283649, −5.37314233472007462278590061308, −5.18058364028652628663023411138, −4.83747016575679997910041884614, −4.61623785738444110451720154282, −4.47140250956091019019732810514, −4.09241325446444194011138237968, −3.97448122760600372419640997376, −3.71528161182424908862997565148, −3.61095917898919363371828962746, −3.56408390987197825658459308347, −2.88145826768297973328942596820, −2.66057316537643582387278013236, −2.54743812765782901692226842624, −2.24637538348886894622869036453, −1.99802727104777737060257644841, −1.79694761016219889223867217409, −1.64349476822250994938637824157, −1.09206074256356133966149196844, −0.74075878314470059071410465677, −0.20810773747728618761577760139, 0.20810773747728618761577760139, 0.74075878314470059071410465677, 1.09206074256356133966149196844, 1.64349476822250994938637824157, 1.79694761016219889223867217409, 1.99802727104777737060257644841, 2.24637538348886894622869036453, 2.54743812765782901692226842624, 2.66057316537643582387278013236, 2.88145826768297973328942596820, 3.56408390987197825658459308347, 3.61095917898919363371828962746, 3.71528161182424908862997565148, 3.97448122760600372419640997376, 4.09241325446444194011138237968, 4.47140250956091019019732810514, 4.61623785738444110451720154282, 4.83747016575679997910041884614, 5.18058364028652628663023411138, 5.37314233472007462278590061308, 5.49159182937081071445558283649, 5.69490244023560750193203742191, 6.15477963843348821495211188053, 6.21119806104241908709911456345, 6.32630323042012411979064576657

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