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Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·4-s + 3·9-s − 6·12-s + 5·16-s + 6·25-s − 4·27-s + 9·36-s − 2·37-s + 12·41-s − 24·47-s − 10·48-s − 14·49-s − 12·53-s + 3·64-s + 24·67-s + 24·71-s + 12·73-s − 12·75-s + 5·81-s − 24·83-s + 18·100-s − 12·101-s − 12·108-s + 4·111-s − 22·121-s − 24·123-s + ⋯
L(s)  = 1  − 1.15·3-s + 3/2·4-s + 9-s − 1.73·12-s + 5/4·16-s + 6/5·25-s − 0.769·27-s + 3/2·36-s − 0.328·37-s + 1.87·41-s − 3.50·47-s − 1.44·48-s − 2·49-s − 1.64·53-s + 3/8·64-s + 2.93·67-s + 2.84·71-s + 1.40·73-s − 1.38·75-s + 5/9·81-s − 2.63·83-s + 9/5·100-s − 1.19·101-s − 1.15·108-s + 0.379·111-s − 2·121-s − 2.16·123-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: \(0\)
Selberg data: \((4,\ 12321,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.021999846\)
\(L(\frac12)\) \(\approx\) \(1.021999846\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
37$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−14.17952310701872949859595250521, −12.87565285401432872639262178272, −12.79787243985353298223709036020, −12.44398550665205743228137762624, −11.42825440010378608142799036587, −11.38146260772251016527345629756, −11.06460693820284492355400534252, −10.46292691465956144092347120015, −9.739559806398530334351747430084, −9.424740030545842095712967049081, −8.104204885157162945716472637486, −7.975106550297768615881604701491, −6.92880119301849570264539567594, −6.56518043014372170255588648685, −6.32673893184291333903604628812, −5.28006451799611875183998717184, −4.89634428026468725894188041316, −3.69096110124290817114033232686, −2.71360488779594670597507151626, −1.54034535913873605036438285380, 1.54034535913873605036438285380, 2.71360488779594670597507151626, 3.69096110124290817114033232686, 4.89634428026468725894188041316, 5.28006451799611875183998717184, 6.32673893184291333903604628812, 6.56518043014372170255588648685, 6.92880119301849570264539567594, 7.975106550297768615881604701491, 8.104204885157162945716472637486, 9.424740030545842095712967049081, 9.739559806398530334351747430084, 10.46292691465956144092347120015, 11.06460693820284492355400534252, 11.38146260772251016527345629756, 11.42825440010378608142799036587, 12.44398550665205743228137762624, 12.79787243985353298223709036020, 12.87565285401432872639262178272, 14.17952310701872949859595250521

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