Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{8} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 7·5-s + 15·7-s − 8·8-s + 14·10-s + 15·11-s − 20·13-s + 30·14-s − 16·16-s + 16·17-s + 30·22-s + 6·23-s + 25·25-s − 40·26-s + 10·29-s − 93·31-s + 32·34-s + 105·35-s − 20·37-s − 56·40-s − 50·41-s − 30·43-s + 12·46-s + 150·47-s + 101·49-s + 50·50-s + 94·53-s + ⋯
L(s)  = 1  + 2-s + 7/5·5-s + 15/7·7-s − 8-s + 7/5·10-s + 1.36·11-s − 1.53·13-s + 15/7·14-s − 16-s + 0.941·17-s + 1.36·22-s + 6/23·23-s + 25-s − 1.53·26-s + 0.344·29-s − 3·31-s + 0.941·34-s + 3·35-s − 0.540·37-s − 7/5·40-s − 1.21·41-s − 0.697·43-s + 6/23·46-s + 3.19·47-s + 2.06·49-s + 50-s + 1.77·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(104976\)    =    \(2^{4} \cdot 3^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{324} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 104976,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(5.73125\)
\(L(\frac12)\)  \(\approx\)  \(5.73125\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - p T + p^{2} T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 7 T + 24 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 13 T + p^{2} T^{2} )( 1 - 2 T + p^{2} T^{2} ) \)
11$C_2^2$ \( 1 - 15 T + 196 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2^2$ \( 1 + 20 T + 231 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 614 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 541 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 10 T - 741 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \)
31$C_1$$\times$$C_2$ \( ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 50 T + 819 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2^2$ \( 1 + 30 T + 2149 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 150 T + 9709 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 47 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 60 T + 4681 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 64 T + 375 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 150 T + 11989 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 55 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 12 T + 6289 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 51 T + 7756 T^{2} - 51 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 25 T - 8784 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \)
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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

−11.98415199368103513079729250517, −11.32899332494845834323240062973, −10.81016606795631989351575136838, −10.13093106706783645956016682100, −9.880064674355905781753290498473, −9.241633686963044103515002561956, −8.736185785850588180198835842882, −8.634370133046057682848018789257, −7.69215276988318674419371619368, −7.08060232345992086063322935674, −6.93504819648759090601672873619, −5.77302663042374100094245905697, −5.63409472026049273549262988985, −5.12364530686031813731415793501, −4.80723172736558676201753911785, −4.00146740709478960464070738973, −3.53350842986239504052124044526, −2.27323817870885253834313826135, −2.02839216935656648810790280521, −1.06553234349302067555067475821, 1.06553234349302067555067475821, 2.02839216935656648810790280521, 2.27323817870885253834313826135, 3.53350842986239504052124044526, 4.00146740709478960464070738973, 4.80723172736558676201753911785, 5.12364530686031813731415793501, 5.63409472026049273549262988985, 5.77302663042374100094245905697, 6.93504819648759090601672873619, 7.08060232345992086063322935674, 7.69215276988318674419371619368, 8.634370133046057682848018789257, 8.736185785850588180198835842882, 9.241633686963044103515002561956, 9.880064674355905781753290498473, 10.13093106706783645956016682100, 10.81016606795631989351575136838, 11.32899332494845834323240062973, 11.98415199368103513079729250517

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