Properties

Label 4-111e2-1.1-c0e2-0-2
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  − 3-s + 4-s − 7-s − 12-s − 3·13-s + 21-s + 25-s + 27-s − 28-s + 2·37-s + 3·39-s + 49-s − 3·52-s − 64-s − 67-s + 2·73-s − 75-s − 3·79-s − 81-s + 84-s + 3·91-s + 100-s + 108-s − 3·109-s − 2·111-s + 2·121-s + 127-s + ⋯
L(s)  = 1  − 3-s + 4-s − 7-s − 12-s − 3·13-s + 21-s + 25-s + 27-s − 28-s + 2·37-s + 3·39-s + 49-s − 3·52-s − 64-s − 67-s + 2·73-s − 75-s − 3·79-s − 81-s + 84-s + 3·91-s + 100-s + 108-s − 3·109-s − 2·111-s + 2·121-s + 127-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

\(L(\frac{1}{2})\) \(\approx\) \(0.2786522884\)
\(L(\frac12)\) \(\approx\) \(0.2786522884\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_2^2$ \( 1 - T^{2} + T^{4} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−14.36833704050682452832322795010, −13.55016410918758895210459647472, −12.72661832174701726040763085903, −12.61525216977882003906043806501, −11.91496355477986837445357335560, −11.79772524475894385983329721976, −11.02828773337518437619578361241, −10.63744028703307335460013897015, −9.896913020367610485625389816851, −9.678190572919910280677778371520, −8.971057301257502278056761614457, −7.997035025629676928244448689468, −7.19538498646848317933120534875, −7.05344265564969174407287336327, −6.35795379858912165737410810501, −5.76006380869294265534927873326, −5.00533503922753385933189223635, −4.41656836049261226463945647116, −2.87499744189534508364905084632, −2.51158045381864168173649537375, 2.51158045381864168173649537375, 2.87499744189534508364905084632, 4.41656836049261226463945647116, 5.00533503922753385933189223635, 5.76006380869294265534927873326, 6.35795379858912165737410810501, 7.05344265564969174407287336327, 7.19538498646848317933120534875, 7.997035025629676928244448689468, 8.971057301257502278056761614457, 9.678190572919910280677778371520, 9.896913020367610485625389816851, 10.63744028703307335460013897015, 11.02828773337518437619578361241, 11.79772524475894385983329721976, 11.91496355477986837445357335560, 12.61525216977882003906043806501, 12.72661832174701726040763085903, 13.55016410918758895210459647472, 14.36833704050682452832322795010

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