Properties

Degree $4$
Conductor $10653696$
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  − 9-s − 4·13-s + 2·17-s − 8·19-s + 10·25-s − 8·43-s + 16·47-s − 2·49-s − 12·53-s − 24·59-s + 24·67-s + 81-s + 24·83-s − 20·89-s + 12·101-s + 4·117-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.10·13-s + 0.485·17-s − 1.83·19-s + 2·25-s − 1.21·43-s + 2.33·47-s − 2/7·49-s − 1.64·53-s − 3.12·59-s + 2.93·67-s + 1/9·81-s + 2.63·83-s − 2.11·89-s + 1.19·101-s + 0.369·117-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10653696\)    =    \(2^{12} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3264} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10653696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.242099520\)
\(L(\frac12)\) \(\approx\) \(1.242099520\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−9.092073775247950372142565643745, −8.504995662132083264407473647956, −8.093918147859020171507176024935, −7.61971616911127348160666198782, −7.54589818118301820357274453582, −6.75928121739465375726192172993, −6.67519169143584120937097545513, −6.31409620129345260214814525694, −5.87322303603784433446890478681, −5.23750171066785105713896496118, −5.07841413722666243729852725725, −4.52526555564160521048747049391, −4.40825594732626851832215490337, −3.57119675450040322613192029112, −3.37721044627627627731585389728, −2.59924538469102992238637666025, −2.50410773825798860514505138230, −1.82518497361076507274090842118, −1.17429122779692212625963207886, −0.35720249118236664948040314366, 0.35720249118236664948040314366, 1.17429122779692212625963207886, 1.82518497361076507274090842118, 2.50410773825798860514505138230, 2.59924538469102992238637666025, 3.37721044627627627731585389728, 3.57119675450040322613192029112, 4.40825594732626851832215490337, 4.52526555564160521048747049391, 5.07841413722666243729852725725, 5.23750171066785105713896496118, 5.87322303603784433446890478681, 6.31409620129345260214814525694, 6.67519169143584120937097545513, 6.75928121739465375726192172993, 7.54589818118301820357274453582, 7.61971616911127348160666198782, 8.093918147859020171507176024935, 8.504995662132083264407473647956, 9.092073775247950372142565643745

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