Properties

Degree $4$
Conductor $15876$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 7·5-s − 20·7-s + 8·8-s − 14·10-s + 35·11-s + 132·13-s + 40·14-s − 16·16-s + 59·17-s − 137·19-s − 70·22-s − 7·23-s + 125·25-s − 264·26-s − 212·29-s − 75·31-s − 118·34-s − 140·35-s − 11·37-s + 274·38-s + 56·40-s + 996·41-s + 520·43-s + 14·46-s − 171·47-s + 57·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.626·5-s − 1.07·7-s + 0.353·8-s − 0.442·10-s + 0.959·11-s + 2.81·13-s + 0.763·14-s − 1/4·16-s + 0.841·17-s − 1.65·19-s − 0.678·22-s − 0.0634·23-s + 25-s − 1.99·26-s − 1.35·29-s − 0.434·31-s − 0.595·34-s − 0.676·35-s − 0.0488·37-s + 1.16·38-s + 0.221·40-s + 3.79·41-s + 1.84·43-s + 0.0448·46-s − 0.530·47-s + 0.166·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15876\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{126} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15876,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63183\)
\(L(\frac12)\) \(\approx\) \(1.63183\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 20 T + p^{3} T^{2} \)
good5$C_2^2$ \( 1 - 7 T - 76 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 35 T - 106 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 66 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 59 T - 1432 T^{2} - 59 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 137 T + 11910 T^{2} + 137 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 7 T - 12118 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 106 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 75 T - 24166 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 11 T - 50532 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 498 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 260 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 171 T - 74582 T^{2} + 171 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 417 T + 25012 T^{2} + 417 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 17 T - 205090 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 51 T - 224380 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 439 T - 108042 T^{2} + 439 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 784 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 295 T - 301992 T^{2} + 295 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 495 T - 248014 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 932 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 873 T + 57160 T^{2} + 873 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 290 T + p^{3} T^{2} )^{2} \)
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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

−13.00579591233581098222925099371, −12.70553890088900396898938867677, −12.43839036002574441614819659114, −11.12775368802122536829384933195, −10.94472754849607374695853521472, −10.85187120397412195963459934162, −9.686843395770688861954252744410, −9.556365322412541724954243749666, −8.937302555350555764990247611080, −8.623291758005816389457599828309, −7.929807785149124856465909414991, −7.15353843643374221300749993991, −6.34742217827225993570876364248, −6.10482947107831492140650999615, −5.65129927973124152435328074401, −4.09579777439409544376853562739, −3.94231990908140349888808183708, −2.89321930069236891371525920164, −1.62684346070702057831138793024, −0.819899118390465931730992080482, 0.819899118390465931730992080482, 1.62684346070702057831138793024, 2.89321930069236891371525920164, 3.94231990908140349888808183708, 4.09579777439409544376853562739, 5.65129927973124152435328074401, 6.10482947107831492140650999615, 6.34742217827225993570876364248, 7.15353843643374221300749993991, 7.929807785149124856465909414991, 8.623291758005816389457599828309, 8.937302555350555764990247611080, 9.556365322412541724954243749666, 9.686843395770688861954252744410, 10.85187120397412195963459934162, 10.94472754849607374695853521472, 11.12775368802122536829384933195, 12.43839036002574441614819659114, 12.70553890088900396898938867677, 13.00579591233581098222925099371

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