Properties

Label 4-142572-1.1-c1e2-0-2
Degree $4$
Conductor $142572$
Sign $1$
Analytic cond. $9.09051$
Root an. cond. $1.73638$
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  + 2·3-s + 4-s + 3·5-s + 7-s + 2·12-s + 6·15-s + 16-s + 3·20-s + 2·21-s − 25-s − 5·27-s + 28-s + 9·29-s + 31-s + 3·35-s + 4·43-s + 2·48-s − 11·49-s + 6·60-s + 13·61-s + 64-s − 9·71-s − 2·73-s − 2·75-s + 3·80-s − 4·81-s − 18·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.577·12-s + 1.54·15-s + 1/4·16-s + 0.670·20-s + 0.436·21-s − 1/5·25-s − 0.962·27-s + 0.188·28-s + 1.67·29-s + 0.179·31-s + 0.507·35-s + 0.609·43-s + 0.288·48-s − 1.57·49-s + 0.774·60-s + 1.66·61-s + 1/8·64-s − 1.06·71-s − 0.234·73-s − 0.230·75-s + 0.335·80-s − 4/9·81-s − 1.97·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(142572\)    =    \(2^{2} \cdot 3 \cdot 109^{2}\)
Sign: $1$
Analytic conductor: \(9.09051\)
Root analytic conductor: \(1.73638\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 142572,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.463191633\)
\(L(\frac12)\) \(\approx\) \(3.463191633\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 - T )( 1 - T + p T^{2} ) \)
109$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
37$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−9.336473098750040850489155655124, −8.806840308673572560730737074186, −8.391899483979675592482755086185, −7.966666320516702555547746942422, −7.54411649259926653044801358487, −6.72286979169083995428434301157, −6.46644206722297564797061396645, −5.75530435121550658840676090845, −5.44730954160383960785821623639, −4.68128576699473337277481189035, −3.99512304643175620632381373108, −3.16487986384095514047126023359, −2.65616251387774940163246897532, −2.13446785778834027600131184348, −1.40436374869107075146223341051, 1.40436374869107075146223341051, 2.13446785778834027600131184348, 2.65616251387774940163246897532, 3.16487986384095514047126023359, 3.99512304643175620632381373108, 4.68128576699473337277481189035, 5.44730954160383960785821623639, 5.75530435121550658840676090845, 6.46644206722297564797061396645, 6.72286979169083995428434301157, 7.54411649259926653044801358487, 7.966666320516702555547746942422, 8.391899483979675592482755086185, 8.806840308673572560730737074186, 9.336473098750040850489155655124

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