Properties

Label 4-111e2-1.1-c0e2-0-0
Degree $4$
Conductor $12321$
Sign $1$
Analytic cond. $0.00306874$
Root an. cond. $0.235364$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 16-s − 4·27-s − 2·37-s + 2·48-s − 2·49-s + 5·81-s + 4·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·3-s + 3·9-s − 16-s − 4·27-s − 2·37-s + 2·48-s − 2·49-s + 5·81-s + 4·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(12321\)    =    \(3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.00306874\)
Root analytic conductor: \(0.235364\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{111} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12321,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2134578211\)
\(L(\frac12)\) \(\approx\) \(0.2134578211\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−13.86662501688781585150006845396, −13.63389415332974561022405771071, −12.88032595670467214222071285001, −12.50712601288678636476174285911, −12.08169828841517647750547962500, −11.37051472087599925796301330751, −11.26336592177830283349310279878, −10.61941457244762634938626630511, −10.09400887276507614513350913030, −9.627286889041242972834937905847, −8.921363424316000325266869365009, −8.109234277758529601641032437684, −7.23298105014718581945665391039, −6.88701971246819184873514860915, −6.28258528956796865322833558727, −5.66648124971551170411397274205, −4.94230393511775675506662434397, −4.53540235685744608690914638694, −3.54574592690469057076498645767, −1.81269192173086389162999101018, 1.81269192173086389162999101018, 3.54574592690469057076498645767, 4.53540235685744608690914638694, 4.94230393511775675506662434397, 5.66648124971551170411397274205, 6.28258528956796865322833558727, 6.88701971246819184873514860915, 7.23298105014718581945665391039, 8.109234277758529601641032437684, 8.921363424316000325266869365009, 9.627286889041242972834937905847, 10.09400887276507614513350913030, 10.61941457244762634938626630511, 11.26336592177830283349310279878, 11.37051472087599925796301330751, 12.08169828841517647750547962500, 12.50712601288678636476174285911, 12.88032595670467214222071285001, 13.63389415332974561022405771071, 13.86662501688781585150006845396

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