Properties

Label 4-570e2-1.1-c1e2-0-1
Degree $4$
Conductor $324900$
Sign $1$
Analytic cond. $20.7159$
Root an. cond. $2.13341$
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-s − 3-s + 5-s + 6-s − 2·7-s + 8-s − 10-s − 6·11-s − 2·13-s + 2·14-s − 15-s − 16-s + 6·17-s − 7·19-s + 2·21-s + 6·22-s − 9·23-s − 24-s + 2·26-s + 27-s + 30-s − 8·31-s + 6·33-s − 6·34-s − 2·35-s + 10·37-s + 7·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 1.80·11-s − 0.554·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 1.60·19-s + 0.436·21-s + 1.27·22-s − 1.87·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.182·30-s − 1.43·31-s + 1.04·33-s − 1.02·34-s − 0.338·35-s + 1.64·37-s + 1.13·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(20.7159\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2868396421\)
\(L(\frac12)\) \(\approx\) \(0.2868396421\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 + 7 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−10.94873816678968340129075164655, −10.16638017014507483516433699708, −10.09003637158575800221049725533, −9.889088692994453425513456718583, −9.402657941259542503965876920269, −8.750039238270802997566335987055, −8.160784941207707021441747817290, −7.993120986924548534762000072924, −7.51857904000476600701679172902, −6.96557143561581609103695248426, −6.38873088312627976819022714305, −5.84590951473238843458065623410, −5.50777839834613963862777190539, −5.21298337763800090190969209635, −4.11896752997089085795338377348, −4.07941694382071114517495335772, −2.83036909704991749660056451745, −2.54302185568816031686730999018, −1.68812285603811603991036572624, −0.34306327853343236205424443068, 0.34306327853343236205424443068, 1.68812285603811603991036572624, 2.54302185568816031686730999018, 2.83036909704991749660056451745, 4.07941694382071114517495335772, 4.11896752997089085795338377348, 5.21298337763800090190969209635, 5.50777839834613963862777190539, 5.84590951473238843458065623410, 6.38873088312627976819022714305, 6.96557143561581609103695248426, 7.51857904000476600701679172902, 7.993120986924548534762000072924, 8.160784941207707021441747817290, 8.750039238270802997566335987055, 9.402657941259542503965876920269, 9.889088692994453425513456718583, 10.09003637158575800221049725533, 10.16638017014507483516433699708, 10.94873816678968340129075164655

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