Properties

Label 4-111e2-1.1-c1e2-0-5
Degree $4$
Conductor $12321$
Sign $1$
Analytic cond. $0.785597$
Root an. cond. $0.941456$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3-s + 4·4-s − 6·5-s + 3·6-s − 7-s − 3·8-s + 18·10-s − 4·12-s − 9·13-s + 3·14-s + 6·15-s + 3·16-s − 6·17-s − 6·19-s − 24·20-s + 21-s + 3·24-s + 19·25-s + 27·26-s + 27-s − 4·28-s − 18·30-s − 6·32-s + 18·34-s + 6·35-s − 10·37-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.577·3-s + 2·4-s − 2.68·5-s + 1.22·6-s − 0.377·7-s − 1.06·8-s + 5.69·10-s − 1.15·12-s − 2.49·13-s + 0.801·14-s + 1.54·15-s + 3/4·16-s − 1.45·17-s − 1.37·19-s − 5.36·20-s + 0.218·21-s + 0.612·24-s + 19/5·25-s + 5.29·26-s + 0.192·27-s − 0.755·28-s − 3.28·30-s − 1.06·32-s + 3.08·34-s + 1.01·35-s − 1.64·37-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(12321\)    =    \(3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.785597\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 12321,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 10 T + p T^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 24 T + 251 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 24 T + 281 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
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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

−12.94988809746636458183299566952, −12.48434739870057257824566414875, −12.21165918698668434162996881177, −11.66306559702870213093798131226, −11.03713929211594532081223599661, −10.92195502142850016346224136573, −9.987262794008242628987798647283, −9.775020382879372427649050473806, −8.824506457122171371991179665324, −8.599597614948697678685665348376, −7.985551136281917579492170905665, −7.58790311086488824377797535584, −6.94887604808944865242968641125, −6.67900542768103785569722484279, −5.16534014516323065010330018291, −4.47852998814285297512193731646, −3.79606960382670372045779972142, −2.52759037219994674746512217344, 0, 0, 2.52759037219994674746512217344, 3.79606960382670372045779972142, 4.47852998814285297512193731646, 5.16534014516323065010330018291, 6.67900542768103785569722484279, 6.94887604808944865242968641125, 7.58790311086488824377797535584, 7.985551136281917579492170905665, 8.599597614948697678685665348376, 8.824506457122171371991179665324, 9.775020382879372427649050473806, 9.987262794008242628987798647283, 10.92195502142850016346224136573, 11.03713929211594532081223599661, 11.66306559702870213093798131226, 12.21165918698668434162996881177, 12.48434739870057257824566414875, 12.94988809746636458183299566952

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