Properties

Label 4-18252-1.1-c1e2-0-2
Degree $4$
Conductor $18252$
Sign $1$
Analytic cond. $1.16376$
Root an. cond. $1.03864$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 9-s − 12-s + 6·13-s + 16-s − 6·25-s − 27-s + 36-s − 6·39-s + 8·43-s − 48-s + 10·49-s + 6·52-s − 4·61-s + 64-s + 6·75-s + 81-s − 6·100-s − 32·103-s − 108-s + 6·117-s − 22·121-s + 127-s − 8·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 1.66·13-s + 1/4·16-s − 6/5·25-s − 0.192·27-s + 1/6·36-s − 0.960·39-s + 1.21·43-s − 0.144·48-s + 10/7·49-s + 0.832·52-s − 0.512·61-s + 1/8·64-s + 0.692·75-s + 1/9·81-s − 3/5·100-s − 3.15·103-s − 0.0962·108-s + 0.554·117-s − 2·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18252\)    =    \(2^{2} \cdot 3^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1.16376\)
Root analytic conductor: \(1.03864\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 18252,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.112282735\)
\(L(\frac12)\) \(\approx\) \(1.112282735\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−10.92591047448404489455298703812, −10.57457186614303466943177704363, −10.06454957650976622915451112660, −9.253220401756145526066424253394, −8.893772573450384569097262787010, −8.041746957490896078081456984786, −7.68916893113167905967774177448, −6.87944639643402688270762012935, −6.34006998241373302303396063951, −5.81225327835322207959592349923, −5.31396272181510219841318127340, −4.17395035154105580618519439241, −3.74904387488260970946838288229, −2.59439103665810412476981586962, −1.37286813188261880910767163756, 1.37286813188261880910767163756, 2.59439103665810412476981586962, 3.74904387488260970946838288229, 4.17395035154105580618519439241, 5.31396272181510219841318127340, 5.81225327835322207959592349923, 6.34006998241373302303396063951, 6.87944639643402688270762012935, 7.68916893113167905967774177448, 8.041746957490896078081456984786, 8.893772573450384569097262787010, 9.253220401756145526066424253394, 10.06454957650976622915451112660, 10.57457186614303466943177704363, 10.92591047448404489455298703812

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