Properties

Label 4-1638e2-1.1-c1e2-0-46
Degree $4$
Conductor $2683044$
Sign $1$
Analytic cond. $171.073$
Root an. cond. $3.61655$
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-s + 5-s − 5·7-s + 8-s − 10-s − 11-s + 2·13-s + 5·14-s − 16-s − 2·17-s − 2·19-s + 22-s + 2·23-s + 5·25-s − 2·26-s − 14·29-s + 3·31-s + 2·34-s − 5·35-s + 2·37-s + 2·38-s + 40-s − 16·41-s − 16·43-s − 2·46-s + 18·49-s − 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s − 1.88·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s + 1.33·14-s − 1/4·16-s − 0.485·17-s − 0.458·19-s + 0.213·22-s + 0.417·23-s + 25-s − 0.392·26-s − 2.59·29-s + 0.538·31-s + 0.342·34-s − 0.845·35-s + 0.328·37-s + 0.324·38-s + 0.158·40-s − 2.49·41-s − 2.43·43-s − 0.294·46-s + 18/7·49-s − 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2683044\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(171.073\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2683044,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−9.052390146672062519661759001090, −8.987502993596546539176358375917, −8.408700540777443383651853930341, −8.321986319479272653051598501084, −7.34857698346096956753519994605, −7.30370645280870270006323951119, −6.69692466510951270840828180251, −6.57924639343688977774191652433, −5.84903467467946763459460985751, −5.79738763322551774832633286997, −5.09291535262411962338791200441, −4.64064294034245620264441927974, −3.99626645461810811324423971507, −3.56448089020997238933474716787, −2.98389922548793269725175542013, −2.79286515478845993797517130510, −1.75906827003116032256639727772, −1.45304953084384183940918788091, 0, 0, 1.45304953084384183940918788091, 1.75906827003116032256639727772, 2.79286515478845993797517130510, 2.98389922548793269725175542013, 3.56448089020997238933474716787, 3.99626645461810811324423971507, 4.64064294034245620264441927974, 5.09291535262411962338791200441, 5.79738763322551774832633286997, 5.84903467467946763459460985751, 6.57924639343688977774191652433, 6.69692466510951270840828180251, 7.30370645280870270006323951119, 7.34857698346096956753519994605, 8.321986319479272653051598501084, 8.408700540777443383651853930341, 8.987502993596546539176358375917, 9.052390146672062519661759001090

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