Properties

Label 4-450468-1.1-c1e2-0-2
Degree $4$
Conductor $450468$
Sign $-1$
Analytic cond. $28.7222$
Root an. cond. $2.31501$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 7-s + 9-s − 12-s + 5·13-s + 16-s − 7·19-s − 21-s − 6·25-s − 27-s + 28-s − 9·31-s + 36-s + 8·37-s − 5·39-s − 10·43-s − 48-s − 11·49-s + 5·52-s + 7·57-s − 4·61-s + 63-s + 64-s − 11·67-s − 11·73-s + 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.288·12-s + 1.38·13-s + 1/4·16-s − 1.60·19-s − 0.218·21-s − 6/5·25-s − 0.192·27-s + 0.188·28-s − 1.61·31-s + 1/6·36-s + 1.31·37-s − 0.800·39-s − 1.52·43-s − 0.144·48-s − 1.57·49-s + 0.693·52-s + 0.927·57-s − 0.512·61-s + 0.125·63-s + 1/8·64-s − 1.34·67-s − 1.28·73-s + 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(450468\)    =    \(2^{2} \cdot 3^{3} \cdot 43 \cdot 97\)
Sign: $-1$
Analytic conductor: \(28.7222\)
Root analytic conductor: \(2.31501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{450468} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 450468,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
43$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 9 T + p T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
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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

−8.289107434048007375836653068385, −7.962135545427101772034705662470, −7.34841652242487393068470721487, −7.00713469668182377262519309883, −6.28029572457690529116543770451, −6.03363719569089360803867664163, −5.84635784287984891609793304761, −4.94264924413944998655443615454, −4.59945208120390450375176657801, −3.86060167625385080512676879859, −3.57147348237821131225127379851, −2.68984528815175237192275465894, −1.82295775951651112617895608911, −1.47031768188170780933779032535, 0, 1.47031768188170780933779032535, 1.82295775951651112617895608911, 2.68984528815175237192275465894, 3.57147348237821131225127379851, 3.86060167625385080512676879859, 4.59945208120390450375176657801, 4.94264924413944998655443615454, 5.84635784287984891609793304761, 6.03363719569089360803867664163, 6.28029572457690529116543770451, 7.00713469668182377262519309883, 7.34841652242487393068470721487, 7.962135545427101772034705662470, 8.289107434048007375836653068385

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