Properties

Label 4-111e2-1.1-c6e2-0-0
Degree $4$
Conductor $12321$
Sign $1$
Analytic cond. $652.087$
Root an. cond. $5.05331$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 54·3-s − 128·4-s + 572·7-s + 2.18e3·9-s + 6.91e3·12-s + 1.22e4·16-s − 3.08e4·21-s − 3.12e4·25-s − 7.87e4·27-s − 7.32e4·28-s − 2.79e5·36-s − 8.92e4·37-s − 6.63e5·48-s + 1.00e4·49-s + 1.25e6·63-s − 1.04e6·64-s − 3.45e5·67-s − 1.27e6·73-s + 1.68e6·75-s + 2.65e6·81-s + 3.95e6·84-s + 4.00e6·100-s + 1.00e7·108-s + 4.81e6·111-s + 7.02e6·112-s + 3.54e6·121-s + 127-s + ⋯
L(s)  = 1  − 2·3-s − 2·4-s + 1.66·7-s + 3·9-s + 4·12-s + 3·16-s − 3.33·21-s − 2·25-s − 4·27-s − 3.33·28-s − 6·36-s − 1.76·37-s − 6·48-s + 0.0857·49-s + 5.00·63-s − 4·64-s − 1.14·67-s − 3.28·73-s + 4·75-s + 5·81-s + 6.67·84-s + 4·100-s + 8·108-s + 3.52·111-s + 5.00·112-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12321\)    =    \(3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(652.087\)
Root analytic conductor: \(5.05331\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12321,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.01703577703\)
\(L(\frac12)\) \(\approx\) \(0.01703577703\)
\(L(4)\) 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)$
bad3$C_1$ \( ( 1 + p^{3} T )^{2} \)
37$C_2$ \( 1 + 89206 T + p^{6} T^{2} \)
good2$C_2$ \( ( 1 + p^{6} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{6} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 286 T + p^{6} T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
13$C_2$ \( ( 1 - 506 T + p^{6} T^{2} )( 1 + 506 T + p^{6} T^{2} ) \)
17$C_2$ \( ( 1 + p^{6} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 10582 T + p^{6} T^{2} )( 1 + 10582 T + p^{6} T^{2} ) \)
23$C_2$ \( ( 1 + p^{6} T^{2} )^{2} \)
29$C_2$ \( ( 1 + p^{6} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 35282 T + p^{6} T^{2} )( 1 + 35282 T + p^{6} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
43$C_2$ \( ( 1 - 111386 T + p^{6} T^{2} )( 1 + 111386 T + p^{6} T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
59$C_2$ \( ( 1 + p^{6} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 420838 T + p^{6} T^{2} )( 1 + 420838 T + p^{6} T^{2} ) \)
67$C_2$ \( ( 1 + 172874 T + p^{6} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
73$C_2$ \( ( 1 + 638066 T + p^{6} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 204622 T + p^{6} T^{2} )( 1 + 204622 T + p^{6} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
89$C_2$ \( ( 1 + p^{6} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 56446 T + p^{6} T^{2} )( 1 + 56446 T + p^{6} T^{2} ) \)
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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

−12.74149505607634014097489902206, −12.04872002053640574674478035087, −11.79535197124107692540523741490, −11.26647220182045276710671022436, −10.62823939702725203819799879523, −10.06970675194574214032241167825, −9.846842742661600171425946846950, −8.963576939211505379175421192757, −8.489231148716900924482129899908, −7.56685659030299104179612212514, −7.55757125053735460451172960184, −6.23569678425618004060296829042, −5.73067644049283814632346067803, −5.10936680907585818678092660409, −4.86391837551363464762479150776, −4.22343227369228464793085277841, −3.73331032317324191730009946592, −1.61949609990589586088283223184, −1.26575634974104731693913199265, −0.05824409940238865133428861955, 0.05824409940238865133428861955, 1.26575634974104731693913199265, 1.61949609990589586088283223184, 3.73331032317324191730009946592, 4.22343227369228464793085277841, 4.86391837551363464762479150776, 5.10936680907585818678092660409, 5.73067644049283814632346067803, 6.23569678425618004060296829042, 7.55757125053735460451172960184, 7.56685659030299104179612212514, 8.489231148716900924482129899908, 8.963576939211505379175421192757, 9.846842742661600171425946846950, 10.06970675194574214032241167825, 10.62823939702725203819799879523, 11.26647220182045276710671022436, 11.79535197124107692540523741490, 12.04872002053640574674478035087, 12.74149505607634014097489902206

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