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Properties

Label 4-6384e2-1.1-c1e2-0-3
Degree $4$
Conductor $40755456$
Sign $1$
Analytic cond. $2598.60$
Root an. cond. $7.13978$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s − 2·7-s + 3·9-s + 4·11-s + 6·13-s − 4·15-s − 4·17-s + 2·19-s − 4·21-s − 2·23-s − 2·25-s + 4·27-s + 8·29-s − 2·31-s + 8·33-s + 4·35-s + 6·37-s + 12·39-s + 4·41-s − 12·43-s − 6·45-s − 10·47-s + 3·49-s − 8·51-s + 8·53-s − 8·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 1.20·11-s + 1.66·13-s − 1.03·15-s − 0.970·17-s + 0.458·19-s − 0.872·21-s − 0.417·23-s − 2/5·25-s + 0.769·27-s + 1.48·29-s − 0.359·31-s + 1.39·33-s + 0.676·35-s + 0.986·37-s + 1.92·39-s + 0.624·41-s − 1.82·43-s − 0.894·45-s − 1.45·47-s + 3/7·49-s − 1.12·51-s + 1.09·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40755456\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2598.60\)
Root analytic conductor: \(7.13978\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40755456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.402532466\)
\(L(\frac12)\) \(\approx\) \(4.402532466\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_g
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_be
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_s
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_c
29$C_4$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_cc
31$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_s
37$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_da
41$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_g
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dy
47$D_{4}$ \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_ek
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dy
59$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_cc
61$C_4$ \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.61.aq_gk
67$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_cg
71$D_{4}$ \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_io
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_bi
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_4$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_fe
97$D_{4}$ \( 1 - 12 T + 210 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_ic
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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

−8.229600518773975382325719341728, −8.101064705661492960728348788384, −7.48463571436766336454265916432, −7.18560899819909786779018688234, −6.71689960474389991312815162106, −6.60983200478485493503080977723, −6.22721386571719066141736222307, −5.90045592668040909303708598680, −5.31806714765456964718011230107, −4.86847347042519147609834628265, −4.21285168566768010393903353135, −4.16079531346818172001126513973, −3.77555050148179383616424215096, −3.52865102916761077414045316887, −2.88878999060875997186142293677, −2.88803127154916857097334783006, −1.93606759727071188688028715989, −1.76341941261367403864714751245, −0.996482657996319566815008687890, −0.56569870646925842625266256142, 0.56569870646925842625266256142, 0.996482657996319566815008687890, 1.76341941261367403864714751245, 1.93606759727071188688028715989, 2.88803127154916857097334783006, 2.88878999060875997186142293677, 3.52865102916761077414045316887, 3.77555050148179383616424215096, 4.16079531346818172001126513973, 4.21285168566768010393903353135, 4.86847347042519147609834628265, 5.31806714765456964718011230107, 5.90045592668040909303708598680, 6.22721386571719066141736222307, 6.60983200478485493503080977723, 6.71689960474389991312815162106, 7.18560899819909786779018688234, 7.48463571436766336454265916432, 8.101064705661492960728348788384, 8.229600518773975382325719341728

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