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Properties

Label 4-111e2-1.1-c1e2-0-4
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·4-s − 2·7-s − 2·9-s − 4·12-s − 8·13-s + 12·16-s + 4·19-s − 2·21-s − 10·25-s − 5·27-s + 8·28-s − 8·31-s + 8·36-s + 2·37-s − 8·39-s + 16·43-s + 12·48-s − 11·49-s + 32·52-s + 4·57-s + 16·61-s + 4·63-s − 32·64-s − 8·67-s + 22·73-s − 10·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s − 0.755·7-s − 2/3·9-s − 1.15·12-s − 2.21·13-s + 3·16-s + 0.917·19-s − 0.436·21-s − 2·25-s − 0.962·27-s + 1.51·28-s − 1.43·31-s + 4/3·36-s + 0.328·37-s − 1.28·39-s + 2.43·43-s + 1.73·48-s − 1.57·49-s + 4.43·52-s + 0.529·57-s + 2.04·61-s + 0.503·63-s − 4·64-s − 0.977·67-s + 2.57·73-s − 1.15·75-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: \(1\)
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$C_2$ \( 1 - T + p T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−11.01260137068449595930875585977, −9.951391238069835801158610265652, −9.819966817497618882768827794755, −9.409913000320422722043609593574, −9.032345851694468357832029856913, −8.280328032649055839192360653081, −7.59911177067371416247669368567, −7.45218161867817824736008552825, −6.07147765498208492889868342360, −5.44973416215471530225317007593, −4.98766701091363209197511690020, −4.04840491645798462404840046787, −3.50910294340479626549928321873, −2.50026843923495696332176907112, 0, 2.50026843923495696332176907112, 3.50910294340479626549928321873, 4.04840491645798462404840046787, 4.98766701091363209197511690020, 5.44973416215471530225317007593, 6.07147765498208492889868342360, 7.45218161867817824736008552825, 7.59911177067371416247669368567, 8.280328032649055839192360653081, 9.032345851694468357832029856913, 9.409913000320422722043609593574, 9.819966817497618882768827794755, 9.951391238069835801158610265652, 11.01260137068449595930875585977

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