Properties

Label 8-1520e4-1.1-c1e4-0-7
Degree $8$
Conductor $5.338\times 10^{12}$
Sign $1$
Analytic cond. $21701.1$
Root an. cond. $3.48385$
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  + 6·9-s − 4·19-s + 8·25-s − 4·29-s − 8·31-s + 22·49-s + 52·59-s + 8·61-s + 16·71-s + 8·79-s + 17·81-s + 32·89-s + 24·101-s + 20·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s − 24·171-s + 173-s + ⋯
L(s)  = 1  + 2·9-s − 0.917·19-s + 8/5·25-s − 0.742·29-s − 1.43·31-s + 22/7·49-s + 6.76·59-s + 1.02·61-s + 1.89·71-s + 0.900·79-s + 17/9·81-s + 3.39·89-s + 2.38·101-s + 1.91·109-s − 3.63·121-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 + 2.61·169-s − 1.83·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(21701.1\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.554221467\)
\(L(\frac12)\) \(\approx\) \(6.554221467\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$D_4\times C_2$ \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 34 T^{2} + 595 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 54 T^{2} + 1715 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 4 T^{2} + 694 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 128 T^{2} + 7506 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 194 T^{2} + 14995 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 26 T + 285 T^{2} - 26 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 166 T^{2} + 15667 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 34 T^{2} + 7075 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 116 T^{2} + 6774 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.81818591236742531507530799066, −6.53847824451944078476942820375, −6.31443354785152808386557464951, −6.31017275991180559942611282348, −6.02828787343619651952209657033, −5.38197193791193878841825432983, −5.30627092516066515155412615147, −5.17451060186451044927248202297, −5.11127440523543769365541882968, −5.06687546685037456833481733925, −4.27739640598052615362890623464, −4.25801468238873508664717172334, −4.00902587571035455570869971860, −3.80605261575253879895754112899, −3.67564156476820274354921270460, −3.65674990583320570145715394290, −2.98928491061079411314799678318, −2.55503064524296876872992391507, −2.52657008241130813305408273596, −2.12888077900738100818390995859, −1.91332638754287278901993752509, −1.74327967522851569861427311134, −0.916294492091942959435409000840, −0.900047003055115815618466990919, −0.63338995933067839325469605499, 0.63338995933067839325469605499, 0.900047003055115815618466990919, 0.916294492091942959435409000840, 1.74327967522851569861427311134, 1.91332638754287278901993752509, 2.12888077900738100818390995859, 2.52657008241130813305408273596, 2.55503064524296876872992391507, 2.98928491061079411314799678318, 3.65674990583320570145715394290, 3.67564156476820274354921270460, 3.80605261575253879895754112899, 4.00902587571035455570869971860, 4.25801468238873508664717172334, 4.27739640598052615362890623464, 5.06687546685037456833481733925, 5.11127440523543769365541882968, 5.17451060186451044927248202297, 5.30627092516066515155412615147, 5.38197193791193878841825432983, 6.02828787343619651952209657033, 6.31017275991180559942611282348, 6.31443354785152808386557464951, 6.53847824451944078476942820375, 6.81818591236742531507530799066

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