Properties

Label 4-3240e2-1.1-c0e2-0-7
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $2.61459$
Root an. cond. $1.27160$
Motivic weight $0$
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  − 2·2-s + 3·4-s − 5-s − 4·8-s + 2·10-s + 5·16-s + 2·19-s − 3·20-s + 23-s − 3·31-s − 6·32-s − 4·38-s + 4·40-s − 2·46-s + 2·47-s + 49-s + 2·53-s + 3·61-s + 6·62-s + 7·64-s + 6·76-s − 3·79-s − 5·80-s + 3·83-s + 3·92-s − 4·94-s − 2·95-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 5-s − 4·8-s + 2·10-s + 5·16-s + 2·19-s − 3·20-s + 23-s − 3·31-s − 6·32-s − 4·38-s + 4·40-s − 2·46-s + 2·47-s + 49-s + 2·53-s + 3·61-s + 6·62-s + 7·64-s + 6·76-s − 3·79-s − 5·80-s + 3·83-s + 3·92-s − 4·94-s − 2·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.61459\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4833150574\)
\(L(\frac12)\) \(\approx\) \(0.4833150574\)
\(L(1)\) 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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 - T^{2} + T^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )^{2} \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−8.892129304618549294684432002935, −8.795877625995565746022054425171, −8.322253063691103526232715978680, −7.937226770852492521139951155570, −7.46180425083176276816000841178, −7.27912523533879543231904159018, −7.00888502137491555693564544509, −6.97886813211644625743582875593, −5.98611218435903779852507887347, −5.76722528973512968031969162656, −5.35751382373342910778605412372, −5.11847098845601104547859308480, −4.00060221469824650592778499589, −3.85773602517784727052389470813, −3.34875778136731301969430803148, −2.94601141062039916783181935240, −2.30070042155526006335616633926, −1.95272012553594927667455717285, −1.07774004749337072189208513184, −0.70540924740832342775693901353, 0.70540924740832342775693901353, 1.07774004749337072189208513184, 1.95272012553594927667455717285, 2.30070042155526006335616633926, 2.94601141062039916783181935240, 3.34875778136731301969430803148, 3.85773602517784727052389470813, 4.00060221469824650592778499589, 5.11847098845601104547859308480, 5.35751382373342910778605412372, 5.76722528973512968031969162656, 5.98611218435903779852507887347, 6.97886813211644625743582875593, 7.00888502137491555693564544509, 7.27912523533879543231904159018, 7.46180425083176276816000841178, 7.937226770852492521139951155570, 8.322253063691103526232715978680, 8.795877625995565746022054425171, 8.892129304618549294684432002935

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