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Properties

Label 4-792e2-1.1-c1e2-0-19
Degree $4$
Conductor $627264$
Sign $1$
Analytic cond. $39.9948$
Root an. cond. $2.51478$
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  − 4·11-s + 6·25-s + 8·31-s − 4·37-s − 8·41-s − 2·49-s + 8·67-s + 8·83-s + 4·97-s − 8·101-s − 8·103-s + 24·107-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 1.20·11-s + 6/5·25-s + 1.43·31-s − 0.657·37-s − 1.24·41-s − 2/7·49-s + 0.977·67-s + 0.878·83-s + 0.406·97-s − 0.796·101-s − 0.788·103-s + 2.32·107-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627264\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(39.9948\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.546252806\)
\(L(\frac12)\) \(\approx\) \(1.546252806\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ai_ck
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.e_ck
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.i_ck
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.43.a_k
47$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.47.a_aba
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.53.a_cw
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.61.a_be
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ai_di
71$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.71.a_w
73$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.73.a_as
79$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.79.a_c
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.ai_eo
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.89.a_de
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ae_fe
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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

−8.485585837447245330936009314330, −7.86625283054298970890470673322, −7.65067612756759688632517302213, −6.91998728268404911649668111344, −6.66858064388274691433360877408, −6.18427977876090445762762845567, −5.50178705111039034559284722557, −5.11076623056258069455196990119, −4.77894185686831238356647618167, −4.16905824040209989087652768054, −3.43058526805238327527110900873, −2.96340895256567703767158843982, −2.42810280321691380344172246602, −1.65464773353732529260119888334, −0.62934390762895055177768515591, 0.62934390762895055177768515591, 1.65464773353732529260119888334, 2.42810280321691380344172246602, 2.96340895256567703767158843982, 3.43058526805238327527110900873, 4.16905824040209989087652768054, 4.77894185686831238356647618167, 5.11076623056258069455196990119, 5.50178705111039034559284722557, 6.18427977876090445762762845567, 6.66858064388274691433360877408, 6.91998728268404911649668111344, 7.65067612756759688632517302213, 7.86625283054298970890470673322, 8.485585837447245330936009314330

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