Properties

Degree $4$
Conductor $44100$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 5-s + 6-s + 4·7-s − 8-s + 10-s + 11-s + 2·13-s + 4·14-s + 15-s − 16-s + 3·19-s + 4·21-s + 22-s − 7·23-s − 24-s + 2·26-s − 27-s − 16·29-s + 30-s + 2·31-s + 33-s + 4·35-s − 11·37-s + 3·38-s + 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.688·19-s + 0.872·21-s + 0.213·22-s − 1.45·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 2.97·29-s + 0.182·30-s + 0.359·31-s + 0.174·33-s + 0.676·35-s − 1.80·37-s + 0.486·38-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64457\)
\(L(\frac12)\) \(\approx\) \(2.64457\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−12.56004300467112020916203357985, −12.09952933732796233662109180171, −11.77263510153502422064628021798, −11.29608300238249074116009719910, −10.66353147211918657398365319474, −10.35341889859694056514130783466, −9.489981253862455401783347380826, −9.221124652520082017090449300598, −8.475475833482913151888050460888, −8.333648896834752917314253732545, −7.44368990681198161912017519393, −7.26011349959651239983017855611, −6.26244529242652534870492501411, −5.57402202169901494428638056400, −5.38337710428824071698166892579, −4.59280485743357610063262082004, −3.79108735080210081194797490015, −3.51196308069530912728472819133, −2.17677101053160040582305696933, −1.68821284009841022998340766378, 1.68821284009841022998340766378, 2.17677101053160040582305696933, 3.51196308069530912728472819133, 3.79108735080210081194797490015, 4.59280485743357610063262082004, 5.38337710428824071698166892579, 5.57402202169901494428638056400, 6.26244529242652534870492501411, 7.26011349959651239983017855611, 7.44368990681198161912017519393, 8.333648896834752917314253732545, 8.475475833482913151888050460888, 9.221124652520082017090449300598, 9.489981253862455401783347380826, 10.35341889859694056514130783466, 10.66353147211918657398365319474, 11.29608300238249074116009719910, 11.77263510153502422064628021798, 12.09952933732796233662109180171, 12.56004300467112020916203357985

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