Properties

Label 4-3120e2-1.1-c1e2-0-9
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $620.673$
Root an. cond. $4.99132$
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  − 4·5-s + 2·7-s − 9-s + 6·13-s + 11·25-s − 8·29-s − 8·35-s − 22·37-s + 4·45-s + 24·47-s − 11·49-s − 14·61-s − 2·63-s − 24·65-s + 8·67-s − 4·73-s + 30·79-s + 81-s + 24·83-s + 12·91-s + 34·97-s + 24·101-s − 6·117-s + 13·121-s − 24·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.755·7-s − 1/3·9-s + 1.66·13-s + 11/5·25-s − 1.48·29-s − 1.35·35-s − 3.61·37-s + 0.596·45-s + 3.50·47-s − 1.57·49-s − 1.79·61-s − 0.251·63-s − 2.97·65-s + 0.977·67-s − 0.468·73-s + 3.37·79-s + 1/9·81-s + 2.63·83-s + 1.25·91-s + 3.45·97-s + 2.38·101-s − 0.554·117-s + 1.18·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(620.673\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.566135408\)
\(L(\frac12)\) \(\approx\) \(1.566135408\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 153 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−8.807790687133701788747884169704, −8.670561353807942853881521158974, −7.924846377542190571649653527470, −7.83932413282090285520971057164, −7.51736672393402651210104398494, −7.21314401508536718566163815545, −6.51467341619914216473512199011, −6.42212443934896954104088443121, −5.84313852738351906219915812401, −5.32864449573469313304043923403, −4.92153114255651728646775539053, −4.75583372827210339527070095391, −3.87664236016544476825820208999, −3.84163152060126039428936225865, −3.41878296046154347191237948652, −3.18073389279516250827162179432, −2.11625828882431194872094760130, −1.88311819840778865908115693326, −1.03288132204173245375016115272, −0.46285307081924519404102932665, 0.46285307081924519404102932665, 1.03288132204173245375016115272, 1.88311819840778865908115693326, 2.11625828882431194872094760130, 3.18073389279516250827162179432, 3.41878296046154347191237948652, 3.84163152060126039428936225865, 3.87664236016544476825820208999, 4.75583372827210339527070095391, 4.92153114255651728646775539053, 5.32864449573469313304043923403, 5.84313852738351906219915812401, 6.42212443934896954104088443121, 6.51467341619914216473512199011, 7.21314401508536718566163815545, 7.51736672393402651210104398494, 7.83932413282090285520971057164, 7.924846377542190571649653527470, 8.670561353807942853881521158974, 8.807790687133701788747884169704

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