Properties

Degree $4$
Conductor $5531904$
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 + 2·5-s + 2·11-s + 8·13-s − 2·15-s + 6·17-s − 8·19-s − 6·23-s + 5·25-s + 27-s − 20·29-s − 4·31-s − 2·33-s − 6·37-s − 8·39-s + 12·41-s − 8·43-s − 8·47-s − 6·51-s − 2·53-s + 4·55-s + 8·57-s + 4·59-s − 8·61-s + 16·65-s − 8·67-s + 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.603·11-s + 2.21·13-s − 0.516·15-s + 1.45·17-s − 1.83·19-s − 1.25·23-s + 25-s + 0.192·27-s − 3.71·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s − 1.28·39-s + 1.87·41-s − 1.21·43-s − 1.16·47-s − 0.840·51-s − 0.274·53-s + 0.539·55-s + 1.05·57-s + 0.520·59-s − 1.02·61-s + 1.98·65-s − 0.977·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.976563458\)
\(L(\frac12)\) \(\approx\) \(1.976563458\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−9.124295105806833773727832379308, −9.026924650502198397311794069882, −8.238293653941586042831929169610, −8.118968934213075201095250999136, −7.82873433281682297105215498016, −7.08913600098806334463234592185, −6.60911955736818105125185209772, −6.53097058987306716963352900002, −5.82846772557277040098980208945, −5.78436641554897193390401523661, −5.52085651025831516810362680221, −4.96992830920037866504649465723, −4.16078704028136684001010309938, −3.92702906281358700852487197379, −3.56394001544549904949620868212, −3.13378328013269600317980961030, −2.14189477527070864479060436619, −1.72401889027666033798574163309, −1.50404947369174956930445069323, −0.49420185314976220282805971217, 0.49420185314976220282805971217, 1.50404947369174956930445069323, 1.72401889027666033798574163309, 2.14189477527070864479060436619, 3.13378328013269600317980961030, 3.56394001544549904949620868212, 3.92702906281358700852487197379, 4.16078704028136684001010309938, 4.96992830920037866504649465723, 5.52085651025831516810362680221, 5.78436641554897193390401523661, 5.82846772557277040098980208945, 6.53097058987306716963352900002, 6.60911955736818105125185209772, 7.08913600098806334463234592185, 7.82873433281682297105215498016, 8.118968934213075201095250999136, 8.238293653941586042831929169610, 9.026924650502198397311794069882, 9.124295105806833773727832379308

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