Properties

Label 4-5610e2-1.1-c1e2-0-8
Degree $4$
Conductor $31472100$
Sign $1$
Analytic cond. $2006.68$
Root an. cond. $6.69298$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 4·10-s − 2·11-s + 6·12-s − 6·13-s − 4·14-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s − 6·19-s − 6·20-s + 4·21-s + 4·22-s − 2·23-s − 8·24-s + 3·25-s + 12·26-s + 4·27-s + 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s + 1.73·12-s − 1.66·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 1.37·19-s − 1.34·20-s + 0.872·21-s + 0.852·22-s − 0.417·23-s − 1.63·24-s + 3/5·25-s + 2.35·26-s + 0.769·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31472100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2006.68\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 31472100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_4$ \( 1 + 18 T + 154 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 10 T + 118 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−7.86354272231428348680156130691, −7.78614953563936162887314874674, −7.39028888963835754331503056706, −7.32778435837653176903781849563, −6.73573753377005523858574591547, −6.45347128193262451852208633446, −5.92943927286570104718926828240, −5.48698972548728584931224410300, −4.75962679947219534153853845031, −4.75601841967884716540506891250, −4.25108878134129653544110239356, −3.86776990723684138904963576849, −3.09865432756061259663080502397, −2.94906274991163531189299227834, −2.45579756765848827196713573258, −2.21667261034562502990386025769, −1.46567973326919740923751247913, −1.21633840861317687266990410239, 0, 0, 1.21633840861317687266990410239, 1.46567973326919740923751247913, 2.21667261034562502990386025769, 2.45579756765848827196713573258, 2.94906274991163531189299227834, 3.09865432756061259663080502397, 3.86776990723684138904963576849, 4.25108878134129653544110239356, 4.75601841967884716540506891250, 4.75962679947219534153853845031, 5.48698972548728584931224410300, 5.92943927286570104718926828240, 6.45347128193262451852208633446, 6.73573753377005523858574591547, 7.32778435837653176903781849563, 7.39028888963835754331503056706, 7.78614953563936162887314874674, 7.86354272231428348680156130691

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