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Properties

Label 4-43904-1.1-c1e2-0-4
Degree $4$
Conductor $43904$
Sign $1$
Analytic cond. $2.79935$
Root an. cond. $1.29349$
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  + 7-s + 2·9-s + 8·23-s − 6·25-s + 4·29-s + 4·37-s + 49-s − 4·53-s + 2·63-s + 24·67-s − 16·71-s − 5·81-s + 8·107-s + 4·109-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s − 6·169-s + ⋯
L(s)  = 1  + 0.377·7-s + 2/3·9-s + 1.66·23-s − 6/5·25-s + 0.742·29-s + 0.657·37-s + 1/7·49-s − 0.549·53-s + 0.251·63-s + 2.93·67-s − 1.89·71-s − 5/9·81-s + 0.773·107-s + 0.383·109-s − 1.12·113-s − 0.545·121-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 + 0.630·161-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43904\)    =    \(2^{7} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(2.79935\)
Root analytic conductor: \(1.29349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 43904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.479234028\)
\(L(\frac12)\) \(\approx\) \(1.479234028\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ai_bu
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ae_bu
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.41.a_be
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.47.a_aco
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.59.a_as
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.83.a_aco
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.89.a_aek
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.97.a_ade
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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

−10.21153972567606006133163866615, −9.654094699996107964187743515989, −9.276950853486320200719383252542, −8.615872896780573684178537844856, −8.131359784741589074750371519071, −7.57167178525341996409762836293, −7.05393749937669474294137599816, −6.54626652354326609342532266340, −5.85440530134450522286067548295, −5.18544778512776051224363649089, −4.62178247274018777005148327946, −4.00703032339570304850982933014, −3.18534496418189824470981855562, −2.29527196011732596467924605502, −1.21195671673507740852473807590, 1.21195671673507740852473807590, 2.29527196011732596467924605502, 3.18534496418189824470981855562, 4.00703032339570304850982933014, 4.62178247274018777005148327946, 5.18544778512776051224363649089, 5.85440530134450522286067548295, 6.54626652354326609342532266340, 7.05393749937669474294137599816, 7.57167178525341996409762836293, 8.131359784741589074750371519071, 8.615872896780573684178537844856, 9.276950853486320200719383252542, 9.654094699996107964187743515989, 10.21153972567606006133163866615

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