Properties

Degree $4$
Conductor $614656$
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 − 3·5-s + 3·9-s + 3·11-s + 3·15-s + 9·17-s − 7·19-s − 15·23-s + 25-s − 8·27-s − 12·29-s − 5·31-s − 3·33-s + 5·37-s − 9·45-s − 3·47-s − 9·51-s + 9·53-s − 9·55-s + 7·57-s − 9·59-s − 15·61-s − 9·67-s + 15·69-s − 3·73-s − 75-s + 9·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 9-s + 0.904·11-s + 0.774·15-s + 2.18·17-s − 1.60·19-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 + 0.821·37-s − 1.34·45-s − 0.437·47-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 − 1.09·67-s + 1.80·69-s − 0.351·73-s − 0.115·75-s + 1.01·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666990\)
\(L(\frac12)\) \(\approx\) \(0.666990\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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

−10.61134896921866550748012600908, −9.948182265675025666035636128605, −9.903381828491292382157415405779, −9.113931680129063772373857431901, −9.044780945958130956767712445732, −7.980563262373285162574847641449, −7.894984009004472794612556077037, −7.57132766433032385522981738611, −7.35402746501749637698505026235, −6.30631650614521880289639448094, −6.23184523140996318089149251445, −5.76340731267291607594274070710, −5.21034837874184783087873264824, −4.25488960492456462907084354138, −4.21309182963033013230675890583, −3.65090616517273560980214137767, −3.43706915351070291503683370059, −1.87760244114622577592496224355, −1.83573245524652134814498223316, −0.42420496715867480472003911201, 0.42420496715867480472003911201, 1.83573245524652134814498223316, 1.87760244114622577592496224355, 3.43706915351070291503683370059, 3.65090616517273560980214137767, 4.21309182963033013230675890583, 4.25488960492456462907084354138, 5.21034837874184783087873264824, 5.76340731267291607594274070710, 6.23184523140996318089149251445, 6.30631650614521880289639448094, 7.35402746501749637698505026235, 7.57132766433032385522981738611, 7.894984009004472794612556077037, 7.980563262373285162574847641449, 9.044780945958130956767712445732, 9.113931680129063772373857431901, 9.903381828491292382157415405779, 9.948182265675025666035636128605, 10.61134896921866550748012600908

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