Properties

Label 4-6762e2-1.1-c1e2-0-1
Degree $4$
Conductor $45724644$
Sign $1$
Analytic cond. $2915.44$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s + 5-s + 4·6-s − 4·8-s + 3·9-s − 2·10-s − 6·12-s − 5·13-s − 2·15-s + 5·16-s − 6·18-s − 2·19-s + 3·20-s − 2·23-s + 8·24-s + 25-s + 10·26-s − 4·27-s + 7·29-s + 4·30-s − 12·31-s − 6·32-s + 9·36-s + 37-s + 4·38-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 1.41·8-s + 9-s − 0.632·10-s − 1.73·12-s − 1.38·13-s − 0.516·15-s + 5/4·16-s − 1.41·18-s − 0.458·19-s + 0.670·20-s − 0.417·23-s + 1.63·24-s + 1/5·25-s + 1.96·26-s − 0.769·27-s + 1.29·29-s + 0.730·30-s − 2.15·31-s − 1.06·32-s + 3/2·36-s + 0.164·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(45724644\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2915.44\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 45724644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6033429073\)
\(L(\frac12)\) \(\approx\) \(0.6033429073\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 82 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 18 T + 206 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 218 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 17 T + 174 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−8.104604702586528655906445633738, −7.83134093719014011538209227947, −7.29569932191362623130044007211, −7.25186462971707864819415288070, −6.71499611324612220429945360873, −6.61567877628310367378312998863, −6.07756409692961215654746137915, −5.85695241198848078830704039583, −5.32136237583507011365272812964, −5.14341383306162445390040615121, −4.73865508183305681634121369341, −4.26183198850873704387624631815, −3.72988250992142800093473741792, −3.34252058310571327078580677716, −2.63960247895371269782993674535, −2.37242954541483000747713821482, −1.80446741879818055301256070890, −1.57682909990166113307177242521, −0.71510177643000334836138940405, −0.37753831988603133028920515211, 0.37753831988603133028920515211, 0.71510177643000334836138940405, 1.57682909990166113307177242521, 1.80446741879818055301256070890, 2.37242954541483000747713821482, 2.63960247895371269782993674535, 3.34252058310571327078580677716, 3.72988250992142800093473741792, 4.26183198850873704387624631815, 4.73865508183305681634121369341, 5.14341383306162445390040615121, 5.32136237583507011365272812964, 5.85695241198848078830704039583, 6.07756409692961215654746137915, 6.61567877628310367378312998863, 6.71499611324612220429945360873, 7.25186462971707864819415288070, 7.29569932191362623130044007211, 7.83134093719014011538209227947, 8.104604702586528655906445633738

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