Properties

Label 4-632448-1.1-c1e2-0-2
Degree $4$
Conductor $632448$
Sign $1$
Analytic cond. $40.3254$
Root an. cond. $2.51996$
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  − 5·5-s + 5·13-s + 11·17-s + 11·25-s + 12·37-s + 4·49-s + 16·53-s − 6·61-s − 25·65-s + 17·73-s − 55·85-s + 9·89-s − 20·97-s − 12·101-s − 109-s − 18·113-s + 9·121-s − 10·125-s + ⋯
L(s)  = 1  − 2.23·5-s + 1.38·13-s + 2.66·17-s + 11/5·25-s + 1.97·37-s + 4/7·49-s + 2.19·53-s − 0.768·61-s − 3.10·65-s + 1.98·73-s − 5.96·85-s + 0.953·89-s − 2.03·97-s − 1.19·101-s − 0.0957·109-s − 1.69·113-s + 9/11·121-s − 0.894·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 632448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 632448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(632448\)    =    \(2^{7} \cdot 3^{4} \cdot 61\)
Sign: $1$
Analytic conductor: \(40.3254\)
Root analytic conductor: \(2.51996\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 632448,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.464181428\)
\(L(\frac12)\) \(\approx\) \(1.464181428\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
61$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 5 T + p T^{2} ) \)
good5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.f_o
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.11.a_aj
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.13.af_be
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.17.al_ck
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.23.a_am
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.31.a_abg
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.37.am_cw
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.43.a_abd
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.53.aq_gf
59$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.59.a_adm
67$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \) 2.67.a_adr
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.71.a_be
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.73.ar_ii
79$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.79.a_bo
83$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.83.a_ada
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.aj_gw
97$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.u_kz
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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

−8.137659575052270877932245678459, −7.998409487457346370538540129181, −7.66362068702457521636602229446, −7.17912954575759471858606312884, −6.73271090764740978013471138210, −5.99879458607903185282354172557, −5.66831586811699292129897949662, −5.16706021969573664214905197506, −4.35478346070911487799555649048, −4.02480642005170922216980275647, −3.59636431350740082526672524686, −3.26893732357952950363545084732, −2.54692685768961053181408269433, −1.24169015182430041799728977222, −0.73045008236886624698924575794, 0.73045008236886624698924575794, 1.24169015182430041799728977222, 2.54692685768961053181408269433, 3.26893732357952950363545084732, 3.59636431350740082526672524686, 4.02480642005170922216980275647, 4.35478346070911487799555649048, 5.16706021969573664214905197506, 5.66831586811699292129897949662, 5.99879458607903185282354172557, 6.73271090764740978013471138210, 7.17912954575759471858606312884, 7.66362068702457521636602229446, 7.998409487457346370538540129181, 8.137659575052270877932245678459

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