↑

Properties

Label 4-4800e2-1.1-c1e2-0-7
Degree $4$
Conductor $23040000$
Sign $1$
Analytic cond. $1469.05$
Root an. cond. $6.19097$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 8·11-s − 8·19-s − 4·29-s − 16·31-s − 12·41-s + 14·49-s + 24·59-s − 28·61-s + 16·71-s + 16·79-s + 81-s − 20·89-s − 8·99-s − 12·101-s − 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 2.41·11-s − 1.83·19-s − 0.742·29-s − 2.87·31-s − 1.87·41-s + 2·49-s + 3.12·59-s − 3.58·61-s + 1.89·71-s + 1.80·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s − 1.19·101-s − 3.44·109-s + 2.36·121-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 − 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23040000\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1469.05\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23040000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.164407624\)
\(L(\frac12)\) \(\approx\) \(1.164407624\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.43.a_cg
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.61.bc_mg
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.918197958112961084216394547938, −8.088078085062497659881020821369, −7.88172147093439417105074732699, −7.16042493093439203673160904572, −6.95733613986660732011496397802, −6.61548065550678293916053145047, −6.55491585043535998823067326448, −5.76204000281470405600526617975, −5.66910998952363233241607107513, −5.31309530311752212479735715925, −4.65328193659855710671160174356, −4.24464911594140542575326212724, −3.89857759123985928834040086071, −3.60445435903543290321774038593, −3.40882284369630044140144217101, −2.33968411710848432369505541757, −2.29414568521590372842859477024, −1.50794429003346791380644018311, −1.35172418208740981986415162107, −0.28990047752366773298050040821, 0.28990047752366773298050040821, 1.35172418208740981986415162107, 1.50794429003346791380644018311, 2.29414568521590372842859477024, 2.33968411710848432369505541757, 3.40882284369630044140144217101, 3.60445435903543290321774038593, 3.89857759123985928834040086071, 4.24464911594140542575326212724, 4.65328193659855710671160174356, 5.31309530311752212479735715925, 5.66910998952363233241607107513, 5.76204000281470405600526617975, 6.55491585043535998823067326448, 6.61548065550678293916053145047, 6.95733613986660732011496397802, 7.16042493093439203673160904572, 7.88172147093439417105074732699, 8.088078085062497659881020821369, 8.918197958112961084216394547938

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