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Properties

Label 4-111e2-1.1-c1e2-0-2
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  − 3·3-s − 2·7-s + 6·9-s − 4·13-s − 4·16-s + 6·21-s − 6·25-s − 9·27-s − 8·31-s − 2·37-s + 12·39-s + 4·43-s + 12·48-s − 11·49-s − 16·61-s − 12·63-s + 16·67-s − 2·73-s + 18·75-s + 8·79-s + 9·81-s + 8·91-s + 24·93-s + 8·97-s + 36·103-s − 32·109-s + 6·111-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.755·7-s + 2·9-s − 1.10·13-s − 16-s + 1.30·21-s − 6/5·25-s − 1.73·27-s − 1.43·31-s − 0.328·37-s + 1.92·39-s + 0.609·43-s + 1.73·48-s − 1.57·49-s − 2.04·61-s − 1.51·63-s + 1.95·67-s − 0.234·73-s + 2.07·75-s + 0.900·79-s + 81-s + 0.838·91-s + 2.48·93-s + 0.812·97-s + 3.54·103-s − 3.06·109-s + 0.569·111-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$C_2$ \( 1 + p T + p T^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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

−10.92682700120020504050380544848, −10.77513816254080044575148887495, −9.933098353605351714295524224830, −9.565522888261174989247372778120, −9.121146862459146776714056017159, −8.014330807872879223418899196928, −7.33975387724698509253296068252, −6.87039121695443194852465216240, −6.26782702715824593155536544660, −5.72006928538748908141093650774, −5.00317001400665869534627315571, −4.45853218893206174814223169396, −3.48216880696490439128154579636, −2.06170899722132089797222638055, 0, 2.06170899722132089797222638055, 3.48216880696490439128154579636, 4.45853218893206174814223169396, 5.00317001400665869534627315571, 5.72006928538748908141093650774, 6.26782702715824593155536544660, 6.87039121695443194852465216240, 7.33975387724698509253296068252, 8.014330807872879223418899196928, 9.121146862459146776714056017159, 9.565522888261174989247372778120, 9.933098353605351714295524224830, 10.77513816254080044575148887495, 10.92682700120020504050380544848

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