Properties

Label 8-2450e4-1.1-c1e4-0-2
Degree $8$
Conductor $3.603\times 10^{13}$
Sign $1$
Analytic cond. $146478.$
Root an. cond. $4.42304$
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  − 2·4-s + 8·11-s + 3·16-s − 8·19-s + 8·29-s + 16·41-s − 16·44-s − 40·59-s − 8·61-s − 4·64-s + 16·71-s + 16·76-s + 16·79-s + 14·81-s + 8·101-s + 24·109-s − 16·116-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 32·164-s + ⋯
L(s)  = 1  − 4-s + 2.41·11-s + 3/4·16-s − 1.83·19-s + 1.48·29-s + 2.49·41-s − 2.41·44-s − 5.20·59-s − 1.02·61-s − 1/2·64-s + 1.89·71-s + 1.83·76-s + 1.80·79-s + 14/9·81-s + 0.796·101-s + 2.29·109-s − 1.48·116-s + 1.09·121-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 − 2.49·164-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(146478.\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.016166958\)
\(L(\frac12)\) \(\approx\) \(2.016166958\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
11$D_{4}$ \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 44 T^{2} + 1030 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 44 T^{2} + 2854 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 60 T^{2} + 5366 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 172 T^{2} + 14326 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 160 T^{2} + 16546 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 288 T^{2} + 34226 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 64 T^{2} + 9474 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.39399130804145172025830358645, −6.09143497799079737905451503493, −6.01108546143869933971750318286, −5.98611343077453752250912950188, −5.56335605917819474829950019010, −5.10599023679728606407900167558, −5.01702092665671347202660439597, −4.89893057642734328137393798047, −4.52926207254757590896453942987, −4.38935363964853185591959538580, −4.35300909641337723094536001973, −4.08159495995994266439823254636, −3.79276227776567919149712200088, −3.60473933700510773234109545827, −3.44368444177730901346324378500, −3.12584128703692414453406016990, −2.77465960323229942562283565960, −2.68033962629407582682350628222, −2.17954991242100634017009984744, −1.94693181847744733460579298353, −1.77503131842460740954774796660, −1.20516095868115670357061344905, −1.14694028055922218507332079025, −0.77166113878200625449024436728, −0.26729954105367526078283590755, 0.26729954105367526078283590755, 0.77166113878200625449024436728, 1.14694028055922218507332079025, 1.20516095868115670357061344905, 1.77503131842460740954774796660, 1.94693181847744733460579298353, 2.17954991242100634017009984744, 2.68033962629407582682350628222, 2.77465960323229942562283565960, 3.12584128703692414453406016990, 3.44368444177730901346324378500, 3.60473933700510773234109545827, 3.79276227776567919149712200088, 4.08159495995994266439823254636, 4.35300909641337723094536001973, 4.38935363964853185591959538580, 4.52926207254757590896453942987, 4.89893057642734328137393798047, 5.01702092665671347202660439597, 5.10599023679728606407900167558, 5.56335605917819474829950019010, 5.98611343077453752250912950188, 6.01108546143869933971750318286, 6.09143497799079737905451503493, 6.39399130804145172025830358645

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