Properties

Label 4-960e2-1.1-c1e2-0-9
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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 + 3·9-s − 4·13-s − 4·15-s − 25-s + 4·27-s + 16·31-s − 20·37-s − 8·39-s − 12·41-s − 8·43-s − 6·45-s + 10·49-s − 20·53-s + 8·65-s + 24·67-s − 2·75-s + 32·79-s + 5·81-s − 8·83-s + 20·89-s + 32·93-s + 40·107-s − 40·111-s − 12·117-s + 18·121-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s − 1/5·25-s + 0.769·27-s + 2.87·31-s − 3.28·37-s − 1.28·39-s − 1.87·41-s − 1.21·43-s − 0.894·45-s + 10/7·49-s − 2.74·53-s + 0.992·65-s + 2.93·67-s − 0.230·75-s + 3.60·79-s + 5/9·81-s − 0.878·83-s + 2.11·89-s + 3.31·93-s + 3.86·107-s − 3.79·111-s − 1.10·117-s + 1.63·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053070271\)
\(L(\frac12)\) \(\approx\) \(2.053070271\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−10.26326877023450786333653263014, −9.783710539921746821327911348447, −9.450555190198342643969899541030, −8.858339880160757434447485733079, −8.445683784648627203527148812441, −8.164105679605868503469384397387, −7.937785045555799658915061778684, −7.33069204645065845834476019385, −6.92336434428169285257838689060, −6.63651078436676250147398273762, −6.11422327474390747746602661890, −5.04614897444894229080799760269, −4.98151294746872336781142034815, −4.56993818126751281695178133102, −3.68457356159424420453083251149, −3.45970324347205105948798716800, −3.05618304967124334314918736119, −2.16283326303674544366384497840, −1.86146135728829089669358620016, −0.61129509207960552150946586047, 0.61129509207960552150946586047, 1.86146135728829089669358620016, 2.16283326303674544366384497840, 3.05618304967124334314918736119, 3.45970324347205105948798716800, 3.68457356159424420453083251149, 4.56993818126751281695178133102, 4.98151294746872336781142034815, 5.04614897444894229080799760269, 6.11422327474390747746602661890, 6.63651078436676250147398273762, 6.92336434428169285257838689060, 7.33069204645065845834476019385, 7.937785045555799658915061778684, 8.164105679605868503469384397387, 8.445683784648627203527148812441, 8.858339880160757434447485733079, 9.450555190198342643969899541030, 9.783710539921746821327911348447, 10.26326877023450786333653263014

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