Properties

Degree $4$
Conductor $5531904$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s + 3·9-s + 4·11-s + 8·13-s − 8·15-s + 4·17-s + 4·23-s + 4·25-s − 4·27-s − 8·29-s + 8·31-s − 8·33-s − 8·37-s − 16·39-s + 4·41-s + 12·45-s − 8·51-s − 4·53-s + 16·55-s + 8·59-s + 16·61-s + 32·65-s − 8·69-s + 4·71-s + 8·73-s − 8·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s + 2.21·13-s − 2.06·15-s + 0.970·17-s + 0.834·23-s + 4/5·25-s − 0.769·27-s − 1.48·29-s + 1.43·31-s − 1.39·33-s − 1.31·37-s − 2.56·39-s + 0.624·41-s + 1.78·45-s − 1.12·51-s − 0.549·53-s + 2.15·55-s + 1.04·59-s + 2.04·61-s + 3.96·65-s − 0.963·69-s + 0.474·71-s + 0.936·73-s − 0.923·75-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

\(L(1)\) \(\approx\) \(3.890223476\)
\(L(\frac12)\) \(\approx\) \(3.890223476\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 168 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−9.224747417084806610726881622867, −8.948785680219725786064034058121, −8.401417753965559604637867534714, −8.232461763241298570743706251593, −7.46116002768364918276142653332, −7.07527813623182674604331448633, −6.66206693670730395768220226075, −6.29016947629675909670637238135, −6.09734909584625544952030094694, −5.71067729578087113164558481973, −5.35656433313331858707770433829, −5.15470882676157149658205198511, −4.26832798183318853684641127784, −4.05771268523325580781264186029, −3.42827262935979893026318338962, −3.13331431877583849384499406324, −1.95610814200416690071528689067, −1.93537699519779203874374196514, −1.07548200600560796370249301680, −0.929291434904822367632044584557, 0.929291434904822367632044584557, 1.07548200600560796370249301680, 1.93537699519779203874374196514, 1.95610814200416690071528689067, 3.13331431877583849384499406324, 3.42827262935979893026318338962, 4.05771268523325580781264186029, 4.26832798183318853684641127784, 5.15470882676157149658205198511, 5.35656433313331858707770433829, 5.71067729578087113164558481973, 6.09734909584625544952030094694, 6.29016947629675909670637238135, 6.66206693670730395768220226075, 7.07527813623182674604331448633, 7.46116002768364918276142653332, 8.232461763241298570743706251593, 8.401417753965559604637867534714, 8.948785680219725786064034058121, 9.224747417084806610726881622867

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