Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{4} $
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 + 6·11-s + 4·13-s + 4·19-s + 6·23-s + 5·25-s + 27-s + 12·29-s − 8·31-s − 6·33-s − 2·37-s − 4·39-s + 24·41-s − 8·43-s − 12·47-s + 6·53-s − 4·57-s + 10·61-s − 8·67-s − 6·69-s + 12·71-s + 10·73-s − 5·75-s + 4·79-s − 81-s − 24·83-s − 12·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.80·11-s + 1.10·13-s + 0.917·19-s + 1.25·23-s + 25-s + 0.192·27-s + 2.22·29-s − 1.43·31-s − 1.04·33-s − 0.328·37-s − 0.640·39-s + 3.74·41-s − 1.21·43-s − 1.75·47-s + 0.824·53-s − 0.529·57-s + 1.28·61-s − 0.977·67-s − 0.722·69-s + 1.42·71-s + 1.17·73-s − 0.577·75-s + 0.450·79-s − 1/9·81-s − 2.63·83-s − 1.28·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{588} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 345744,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.05606\)
\(L(\frac12)\)  \(\approx\)  \(2.05606\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.87020691644171622528047026403, −10.79094653225598986408776129999, −9.903872468831631807239936662163, −9.675379819704747705888887866276, −9.055002276238311523165184370499, −8.870941719115200052223813108011, −8.367263437544741754394906904257, −7.87656267761473699156152974313, −6.97806787793626094328148046787, −6.97205489038106508013561606991, −6.37666893824194792476706273193, −6.05518776469725952841377631409, −5.28038697610206779737532920156, −5.06811282590913572284262078589, −4.17534176358488821742811419980, −3.92971438854184200707455607170, −3.16765464597929940935755513166, −2.62376870101628358339758927299, −1.23902767568951116073407019843, −1.11439675822313629821117992368, 1.11439675822313629821117992368, 1.23902767568951116073407019843, 2.62376870101628358339758927299, 3.16765464597929940935755513166, 3.92971438854184200707455607170, 4.17534176358488821742811419980, 5.06811282590913572284262078589, 5.28038697610206779737532920156, 6.05518776469725952841377631409, 6.37666893824194792476706273193, 6.97205489038106508013561606991, 6.97806787793626094328148046787, 7.87656267761473699156152974313, 8.367263437544741754394906904257, 8.870941719115200052223813108011, 9.055002276238311523165184370499, 9.675379819704747705888887866276, 9.903872468831631807239936662163, 10.79094653225598986408776129999, 10.87020691644171622528047026403

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