Properties

Label 4-1280e2-1.1-c1e2-0-32
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
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 + 6·7-s + 4·15-s + 4·19-s − 12·21-s + 6·23-s + 3·25-s + 2·27-s + 12·31-s − 12·35-s + 12·37-s − 12·41-s + 10·43-s + 6·47-s + 16·49-s − 8·57-s + 12·59-s − 12·61-s − 10·67-s − 12·69-s + 12·71-s + 8·73-s − 6·75-s + 24·79-s − 81-s − 6·83-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 2.26·7-s + 1.03·15-s + 0.917·19-s − 2.61·21-s + 1.25·23-s + 3/5·25-s + 0.384·27-s + 2.15·31-s − 2.02·35-s + 1.97·37-s − 1.87·41-s + 1.52·43-s + 0.875·47-s + 16/7·49-s − 1.05·57-s + 1.56·59-s − 1.53·61-s − 1.22·67-s − 1.44·69-s + 1.42·71-s + 0.936·73-s − 0.692·75-s + 2.70·79-s − 1/9·81-s − 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.881460321\)
\(L(\frac12)\) \(\approx\) \(1.881460321\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−9.897584006791258754958413460763, −9.519403402484082955659663881526, −8.793071638074079137227094806959, −8.744804945927095622751562737199, −7.994110742227835553764499171902, −7.916359515231302360463770440660, −7.57468181088586616254724178228, −7.09902113784128903080692335842, −6.34974248156783198313216043871, −6.29159702188483236389418914929, −5.42070708329713475773594404896, −5.25120503777538650817392480543, −4.77445186113098724680085868514, −4.57308852427161751172368002000, −4.00599210164180113253134503962, −3.30890989451535714487564718137, −2.67985413200864269839250476138, −2.04512602556676338221682428014, −1.00700972986514381276153319624, −0.847842980116801684385043312983, 0.847842980116801684385043312983, 1.00700972986514381276153319624, 2.04512602556676338221682428014, 2.67985413200864269839250476138, 3.30890989451535714487564718137, 4.00599210164180113253134503962, 4.57308852427161751172368002000, 4.77445186113098724680085868514, 5.25120503777538650817392480543, 5.42070708329713475773594404896, 6.29159702188483236389418914929, 6.34974248156783198313216043871, 7.09902113784128903080692335842, 7.57468181088586616254724178228, 7.916359515231302360463770440660, 7.994110742227835553764499171902, 8.744804945927095622751562737199, 8.793071638074079137227094806959, 9.519403402484082955659663881526, 9.897584006791258754958413460763

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