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Properties

Label 4-442368-1.1-c1e2-0-16
Degree $4$
Conductor $442368$
Sign $1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
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-s + 2·7-s + 9-s + 4·13-s + 8·19-s − 2·21-s − 2·25-s − 27-s + 2·31-s + 8·37-s − 4·39-s + 12·43-s − 10·49-s − 8·57-s − 8·61-s + 2·63-s + 4·67-s + 12·73-s + 2·75-s + 18·79-s + 81-s + 8·91-s − 2·93-s − 20·97-s + 14·103-s − 4·109-s − 8·111-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 0.436·21-s − 2/5·25-s − 0.192·27-s + 0.359·31-s + 1.31·37-s − 0.640·39-s + 1.82·43-s − 1.42·49-s − 1.05·57-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + 1.40·73-s + 0.230·75-s + 2.02·79-s + 1/9·81-s + 0.838·91-s − 0.207·93-s − 2.03·97-s + 1.37·103-s − 0.383·109-s − 0.759·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.001801645\)
\(L(\frac12)\) \(\approx\) \(2.001801645\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ac_o
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.13.ae_ba
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.23.a_abm
29$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.29.a_abu
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.ac_cc
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ai_cc
41$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.41.a_ak
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.am_eo
47$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.47.a_ba
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.59.a_aec
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.i_fe
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.ae_bm
71$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.71.a_dm
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.am_gk
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.79.as_je
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.u_iw
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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

−8.454473814331367329134444556729, −8.051017571788834912747542741922, −7.68334146859806638809398916303, −7.38058155411024317978703466459, −6.58715794575130066448620490328, −6.27141309034599809760146017112, −5.80383797224955526286135852540, −5.21261506766993869393555765470, −4.95445769558805817996775944144, −4.22361443043057482993898802130, −3.78312884947693409549778140192, −3.12280025830169798817173767961, −2.39398825782297191987730281665, −1.45279728603284071454293255441, −0.902374709471473533403127085790, 0.902374709471473533403127085790, 1.45279728603284071454293255441, 2.39398825782297191987730281665, 3.12280025830169798817173767961, 3.78312884947693409549778140192, 4.22361443043057482993898802130, 4.95445769558805817996775944144, 5.21261506766993869393555765470, 5.80383797224955526286135852540, 6.27141309034599809760146017112, 6.58715794575130066448620490328, 7.38058155411024317978703466459, 7.68334146859806638809398916303, 8.051017571788834912747542741922, 8.454473814331367329134444556729

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