Properties

Degree $4$
Conductor $2286144$
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  + 2·5-s + 5·7-s + 3·11-s + 12·13-s + 4·17-s − 4·19-s − 4·23-s + 5·25-s − 10·29-s − 7·31-s + 10·35-s + 4·41-s + 16·43-s + 2·47-s + 18·49-s − 10·53-s + 6·55-s + 9·59-s + 8·61-s + 24·65-s + 6·67-s − 24·71-s + 11·73-s + 15·77-s − 79-s − 30·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.88·7-s + 0.904·11-s + 3.32·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s + 25-s − 1.85·29-s − 1.25·31-s + 1.69·35-s + 0.624·41-s + 2.43·43-s + 0.291·47-s + 18/7·49-s − 1.37·53-s + 0.809·55-s + 1.17·59-s + 1.02·61-s + 2.97·65-s + 0.733·67-s − 2.84·71-s + 1.28·73-s + 1.70·77-s − 0.112·79-s − 3.29·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.195427535\)
\(L(\frac12)\) \(\approx\) \(5.195427535\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
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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

−9.407306034690403649480154314053, −9.274866866874062377693957655769, −8.769560760090383646019060941791, −8.598866680094166868060650423085, −8.016128353202599543419861539879, −7.977717603118351095639543209294, −7.13987009986289265981839867335, −6.98154507278356048519603531148, −6.17045517261366464361864910677, −5.90508700217059818557946627628, −5.55551756031236371166052003233, −5.52152713495583580465675878186, −4.40946564575686737906665040080, −4.21731716607492629202554147895, −3.79438380996990438978334844847, −3.37790254460282772123371583383, −2.39006247243689420772473708448, −1.85938092054265051136060834018, −1.31548871733038597029413421451, −1.14361214862136589049670968247, 1.14361214862136589049670968247, 1.31548871733038597029413421451, 1.85938092054265051136060834018, 2.39006247243689420772473708448, 3.37790254460282772123371583383, 3.79438380996990438978334844847, 4.21731716607492629202554147895, 4.40946564575686737906665040080, 5.52152713495583580465675878186, 5.55551756031236371166052003233, 5.90508700217059818557946627628, 6.17045517261366464361864910677, 6.98154507278356048519603531148, 7.13987009986289265981839867335, 7.977717603118351095639543209294, 8.016128353202599543419861539879, 8.598866680094166868060650423085, 8.769560760090383646019060941791, 9.274866866874062377693957655769, 9.407306034690403649480154314053

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