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Properties

Label 4-3200e2-1.1-c1e2-0-20
Degree $4$
Conductor $10240000$
Sign $1$
Analytic cond. $652.911$
Root an. cond. $5.05491$
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  + 2·9-s + 4·11-s + 4·19-s − 12·29-s − 12·41-s − 2·49-s + 28·59-s − 4·61-s + 24·71-s + 16·79-s − 5·81-s + 4·89-s + 8·99-s + 12·101-s − 12·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 8·171-s + ⋯
L(s)  = 1  + 2/3·9-s + 1.20·11-s + 0.917·19-s − 2.22·29-s − 1.87·41-s − 2/7·49-s + 3.64·59-s − 0.512·61-s + 2.84·71-s + 1.80·79-s − 5/9·81-s + 0.423·89-s + 0.804·99-s + 1.19·101-s − 1.14·109-s − 0.909·121-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 + 1.69·169-s + 0.611·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10240000\)    =    \(2^{14} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(652.911\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10240000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.019181028\)
\(L(\frac12)\) \(\approx\) \(3.019181028\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
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$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.59.abc_mc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−8.864654347145967582742308943471, −8.512807561853648083021836750402, −8.051934010611822227606673484548, −7.79091872452130419987800747153, −7.17860820859666769606205056802, −7.09616753106336281646677751019, −6.51313928883263793584642993189, −6.50004863402519952311985199909, −5.74263794089906733133104811593, −5.27868198459301103652567821224, −5.23043882596937930279213432593, −4.62293358560158771592294209056, −3.89643697540767564408223287660, −3.84664012265843501797524474657, −3.50027451289608962901077840477, −2.88681064974893808346451669949, −2.01413569000384049387309331000, −1.93293938711150708824877257804, −1.18231914197118704862603796941, −0.58628501449445967857937182606, 0.58628501449445967857937182606, 1.18231914197118704862603796941, 1.93293938711150708824877257804, 2.01413569000384049387309331000, 2.88681064974893808346451669949, 3.50027451289608962901077840477, 3.84664012265843501797524474657, 3.89643697540767564408223287660, 4.62293358560158771592294209056, 5.23043882596937930279213432593, 5.27868198459301103652567821224, 5.74263794089906733133104811593, 6.50004863402519952311985199909, 6.51313928883263793584642993189, 7.09616753106336281646677751019, 7.17860820859666769606205056802, 7.79091872452130419987800747153, 8.051934010611822227606673484548, 8.512807561853648083021836750402, 8.864654347145967582742308943471

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