Properties

Label 4-315e2-1.1-c1e2-0-11
Degree $4$
Conductor $99225$
Sign $1$
Analytic cond. $6.32667$
Root an. cond. $1.58596$
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  + 3·4-s + 5·16-s + 25-s + 20·37-s − 8·43-s − 7·49-s + 3·64-s + 24·67-s + 3·100-s − 28·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 60·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 24·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 3/2·4-s + 5/4·16-s + 1/5·25-s + 3.28·37-s − 1.21·43-s − 49-s + 3/8·64-s + 2.93·67-s + 3/10·100-s − 2.68·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.93·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s − 1.82·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(99225\)    =    \(3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6.32667\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{99225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 99225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.253375518\)
\(L(\frac12)\) \(\approx\) \(2.253375518\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.629316970149713957891799882232, −9.266874953140286712491824843895, −8.318071132202098463163702948204, −8.092721354163082201775927126702, −7.57955991739629067136427002537, −6.98769429632661948164430688998, −6.52212535555073673131226209658, −6.21333713448335166339159279635, −5.55697277880198946381540822310, −4.92294771452158522865499193275, −4.18793092031175921202693687448, −3.44255415723091573070298437865, −2.70558977065123798813507920334, −2.22110289569580257150937635230, −1.20596112310386171313122973365, 1.20596112310386171313122973365, 2.22110289569580257150937635230, 2.70558977065123798813507920334, 3.44255415723091573070298437865, 4.18793092031175921202693687448, 4.92294771452158522865499193275, 5.55697277880198946381540822310, 6.21333713448335166339159279635, 6.52212535555073673131226209658, 6.98769429632661948164430688998, 7.57955991739629067136427002537, 8.092721354163082201775927126702, 8.318071132202098463163702948204, 9.266874953140286712491824843895, 9.629316970149713957891799882232

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