Properties

Degree $4$
Conductor $12348$
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  + 2·2-s + 3·4-s − 7-s + 4·8-s + 9-s − 8·11-s − 2·14-s + 5·16-s + 2·18-s − 16·22-s + 16·23-s − 6·25-s − 3·28-s − 4·29-s + 6·32-s + 3·36-s − 20·37-s − 8·43-s − 24·44-s + 32·46-s + 49-s − 12·50-s + 12·53-s − 4·56-s − 8·58-s − 63-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s + 1/3·9-s − 2.41·11-s − 0.534·14-s + 5/4·16-s + 0.471·18-s − 3.41·22-s + 3.33·23-s − 6/5·25-s − 0.566·28-s − 0.742·29-s + 1.06·32-s + 1/2·36-s − 3.28·37-s − 1.21·43-s − 3.61·44-s + 4.71·46-s + 1/7·49-s − 1.69·50-s + 1.64·53-s − 0.534·56-s − 1.05·58-s − 0.125·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12348\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{12348} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12348,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.071522989\)
\(L(\frac12)\) \(\approx\) \(2.071522989\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−11.37318927677395242391128273111, −10.75802487641203119404618045963, −10.51951977961491945205840276286, −9.933813551342152622597201589196, −9.086022258671443790589095587058, −8.356918966632091922574869118017, −7.68532191797102726979476428936, −6.95470194591461271128583103192, −6.84552480602986516155667793601, −5.47990216984766844713395370376, −5.33985014787602837985094112170, −4.82857955948335988380640999352, −3.62482887081886485478101246558, −3.08481257245522303063597583848, −2.16063348217202322480816158281, 2.16063348217202322480816158281, 3.08481257245522303063597583848, 3.62482887081886485478101246558, 4.82857955948335988380640999352, 5.33985014787602837985094112170, 5.47990216984766844713395370376, 6.84552480602986516155667793601, 6.95470194591461271128583103192, 7.68532191797102726979476428936, 8.356918966632091922574869118017, 9.086022258671443790589095587058, 9.933813551342152622597201589196, 10.51951977961491945205840276286, 10.75802487641203119404618045963, 11.37318927677395242391128273111

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