Properties

Degree $4$
Conductor $22500$
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  − 4-s − 9-s + 16-s + 8·19-s + 12·29-s + 16·31-s + 36-s − 12·41-s − 2·49-s − 20·61-s − 64-s − 8·76-s − 16·79-s + 81-s − 36·89-s + 36·101-s + 20·109-s − 12·116-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s + 1.83·19-s + 2.22·29-s + 2.87·31-s + 1/6·36-s − 1.87·41-s − 2/7·49-s − 2.56·61-s − 1/8·64-s − 0.917·76-s − 1.80·79-s + 1/9·81-s − 3.81·89-s + 3.58·101-s + 1.91·109-s − 1.11·116-s − 2·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(22500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{150} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 22500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07349\)
\(L(\frac12)\) \(\approx\) \(1.07349\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−13.65903418781624892598801003079, −12.73048483344986556804976084426, −12.16301303983113287659365630302, −11.72862706244732673515655909624, −11.57232332102479816450987379595, −10.59711490453538236750892836991, −10.02114576910288854614874100657, −9.941819242818724229118457896505, −9.156664271354851304940002344415, −8.428246467858918435959153272298, −8.303150793602142243576863666858, −7.52893777749673750646095101981, −6.85200004047624573112690645068, −6.24917907500364745105060801431, −5.61952237968355492352832454031, −4.72670202858721634233331620829, −4.56480584669636693327632140720, −3.23242613233288142627718190811, −2.87232562339919378192796826883, −1.19301600866377439702955307792, 1.19301600866377439702955307792, 2.87232562339919378192796826883, 3.23242613233288142627718190811, 4.56480584669636693327632140720, 4.72670202858721634233331620829, 5.61952237968355492352832454031, 6.24917907500364745105060801431, 6.85200004047624573112690645068, 7.52893777749673750646095101981, 8.303150793602142243576863666858, 8.428246467858918435959153272298, 9.156664271354851304940002344415, 9.941819242818724229118457896505, 10.02114576910288854614874100657, 10.59711490453538236750892836991, 11.57232332102479816450987379595, 11.72862706244732673515655909624, 12.16301303983113287659365630302, 12.73048483344986556804976084426, 13.65903418781624892598801003079

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