Properties

Degree $4$
Conductor $12544$
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  − 3-s + 3·5-s + 4·7-s + 3·9-s − 3·11-s − 3·15-s − 9·17-s − 7·19-s − 4·21-s + 15·23-s + 25-s − 8·27-s − 12·29-s − 5·31-s + 3·33-s + 12·35-s + 5·37-s + 9·45-s − 3·47-s + 9·49-s + 9·51-s + 9·53-s − 9·55-s + 7·57-s − 9·59-s + 15·61-s + 12·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 1.51·7-s + 9-s − 0.904·11-s − 0.774·15-s − 2.18·17-s − 1.60·19-s − 0.872·21-s + 3.12·23-s + 1/5·25-s − 1.53·27-s − 2.22·29-s − 0.898·31-s + 0.522·33-s + 2.02·35-s + 0.821·37-s + 1.34·45-s − 0.437·47-s + 9/7·49-s + 1.26·51-s + 1.23·53-s − 1.21·55-s + 0.927·57-s − 1.17·59-s + 1.92·61-s + 1.51·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16723\)
\(L(\frac12)\) \(\approx\) \(1.16723\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 15 T + 98 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 21 T + 236 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−13.48390054678584073455726518985, −13.36435495900951915893436555353, −12.90991557214120478155665202877, −12.74792982753816727685264134953, −11.35668188794049704482501219594, −11.29943998937957979654265389195, −10.77906089588489236258136790061, −10.53881676871788679563498254985, −9.391354172952845164126764313371, −9.343825111569200105677304811287, −8.572229715794336102069115036051, −7.894237687879887047672717842079, −6.99159975970254225762761686382, −6.85205033185553107187980022870, −5.74024338411001776721315514030, −5.35765362414082962117536760423, −4.71525060893812083005003990248, −4.07424526339612746354847297653, −2.29797226428550356109097376842, −1.82698880999477658675413412774, 1.82698880999477658675413412774, 2.29797226428550356109097376842, 4.07424526339612746354847297653, 4.71525060893812083005003990248, 5.35765362414082962117536760423, 5.74024338411001776721315514030, 6.85205033185553107187980022870, 6.99159975970254225762761686382, 7.894237687879887047672717842079, 8.572229715794336102069115036051, 9.343825111569200105677304811287, 9.391354172952845164126764313371, 10.53881676871788679563498254985, 10.77906089588489236258136790061, 11.29943998937957979654265389195, 11.35668188794049704482501219594, 12.74792982753816727685264134953, 12.90991557214120478155665202877, 13.36435495900951915893436555353, 13.48390054678584073455726518985

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