Properties

Degree $4$
Conductor $1806336$
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  − 3-s + 2·5-s + 5·7-s − 6·11-s + 6·13-s − 2·15-s − 4·17-s − 5·19-s − 5·21-s + 4·23-s + 5·25-s + 27-s + 8·29-s − 7·31-s + 6·33-s + 10·35-s − 9·37-s − 6·39-s − 4·41-s + 2·43-s − 2·47-s + 18·49-s + 4·51-s + 8·53-s − 12·55-s + 5·57-s + 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.88·7-s − 1.80·11-s + 1.66·13-s − 0.516·15-s − 0.970·17-s − 1.14·19-s − 1.09·21-s + 0.834·23-s + 25-s + 0.192·27-s + 1.48·29-s − 1.25·31-s + 1.04·33-s + 1.69·35-s − 1.47·37-s − 0.960·39-s − 0.624·41-s + 0.304·43-s − 0.291·47-s + 18/7·49-s + 0.560·51-s + 1.09·53-s − 1.61·55-s + 0.662·57-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.342643325\)
\(L(\frac12)\) \(\approx\) \(2.342643325\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
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$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−10.18644630176911971846820965572, −9.075254067065326305929321751854, −9.046195217543252511145420893041, −8.446274712940039282657322985582, −8.366165907148431719707747166390, −7.995824062294730994288961410975, −7.30084185744788646178363752566, −6.79549591868374692899320467660, −6.68783967896404805240251239016, −5.85875984115795608758901075346, −5.59408299019937659414643583057, −5.27045819999909117227030853843, −4.90310028084826248753660019944, −4.36233391950209015420596156442, −4.02551398396858222818184726107, −3.01464359590303729352082080501, −2.66294329496635965840391789019, −1.78086551505235290708553481262, −1.70312533232172950039168046172, −0.65772956934483450366686268977, 0.65772956934483450366686268977, 1.70312533232172950039168046172, 1.78086551505235290708553481262, 2.66294329496635965840391789019, 3.01464359590303729352082080501, 4.02551398396858222818184726107, 4.36233391950209015420596156442, 4.90310028084826248753660019944, 5.27045819999909117227030853843, 5.59408299019937659414643583057, 5.85875984115795608758901075346, 6.68783967896404805240251239016, 6.79549591868374692899320467660, 7.30084185744788646178363752566, 7.995824062294730994288961410975, 8.366165907148431719707747166390, 8.446274712940039282657322985582, 9.046195217543252511145420893041, 9.075254067065326305929321751854, 10.18644630176911971846820965572

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