Properties

Degree $4$
Conductor $1806336$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s + 7-s − 11-s + 8·13-s + 3·15-s − 4·17-s + 21-s − 8·23-s + 5·25-s − 27-s + 14·29-s − 11·31-s − 33-s + 3·35-s + 4·37-s + 8·39-s − 8·41-s + 4·43-s + 2·47-s − 6·49-s − 4·51-s − 11·53-s − 3·55-s + 7·59-s + 10·61-s + 24·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 0.377·7-s − 0.301·11-s + 2.21·13-s + 0.774·15-s − 0.970·17-s + 0.218·21-s − 1.66·23-s + 25-s − 0.192·27-s + 2.59·29-s − 1.97·31-s − 0.174·33-s + 0.507·35-s + 0.657·37-s + 1.28·39-s − 1.24·41-s + 0.609·43-s + 0.291·47-s − 6/7·49-s − 0.560·51-s − 1.51·53-s − 0.404·55-s + 0.911·59-s + 1.28·61-s + 2.97·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.131525508\)
\(L(\frac12)\) \(\approx\) \(4.131525508\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.838693851739017473414669223754, −9.608586229737677162927271317162, −8.657346263488592570553212705801, −8.613086159089839350282324759769, −8.462824954838264001694209489614, −8.093482024827008278016674666494, −7.17635913379811346826055339069, −7.08683249791056411194129784235, −6.26865157089882679121800009773, −6.09231864265076290503379103225, −5.92284214698367475953356895183, −5.22079413497437144735511355551, −4.72869598950172916959730275789, −4.29059236587322857049971367076, −3.56082050619406454394368842853, −3.37357076077600121728009330204, −2.52015344641385632819445124923, −2.02653109127812968674929359939, −1.68529339660953502136328108613, −0.827964424476160638503126688935, 0.827964424476160638503126688935, 1.68529339660953502136328108613, 2.02653109127812968674929359939, 2.52015344641385632819445124923, 3.37357076077600121728009330204, 3.56082050619406454394368842853, 4.29059236587322857049971367076, 4.72869598950172916959730275789, 5.22079413497437144735511355551, 5.92284214698367475953356895183, 6.09231864265076290503379103225, 6.26865157089882679121800009773, 7.08683249791056411194129784235, 7.17635913379811346826055339069, 8.093482024827008278016674666494, 8.462824954838264001694209489614, 8.613086159089839350282324759769, 8.657346263488592570553212705801, 9.608586229737677162927271317162, 9.838693851739017473414669223754

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