Properties

Label 4-5070e2-1.1-c1e2-0-6
Degree $4$
Conductor $25704900$
Sign $1$
Analytic cond. $1638.96$
Root an. cond. $6.36271$
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  − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 8·23-s − 25-s − 4·27-s − 8·29-s − 3·36-s + 20·43-s − 2·48-s + 5·49-s + 18·53-s − 12·61-s − 64-s − 16·69-s + 2·75-s + 12·79-s + 5·81-s + 16·87-s − 8·92-s + 100-s + 30·103-s + 4·107-s + 4·108-s − 16·113-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 1.66·23-s − 1/5·25-s − 0.769·27-s − 1.48·29-s − 1/2·36-s + 3.04·43-s − 0.288·48-s + 5/7·49-s + 2.47·53-s − 1.53·61-s − 1/8·64-s − 1.92·69-s + 0.230·75-s + 1.35·79-s + 5/9·81-s + 1.71·87-s − 0.834·92-s + 1/10·100-s + 2.95·103-s + 0.386·107-s + 0.384·108-s − 1.50·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25704900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1638.96\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.373046105\)
\(L(\frac12)\) \(\approx\) \(1.373046105\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + T^{2} \)
13 \( 1 \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.601210787736251452283567114581, −7.74376961124057779346326971116, −7.64893602029280037813170848950, −7.36133899707079316986979821576, −7.05417338786111630305243556601, −6.54785454165669105975724525716, −6.08890248327757243666290427977, −5.91226467228514597381191208541, −5.50409982882993187220297422579, −5.08416728227276067570101076998, −4.94550006468922623269185147684, −4.29256231449631530588458288255, −4.07542643680189639343869062064, −3.66066401784555330183746257264, −3.19507717556796983472041937912, −2.43506758899085231630649874501, −2.27518197208888701136112060493, −1.37005452738535426922838548994, −0.957216608346604294854305150230, −0.43778990589117934658639755139, 0.43778990589117934658639755139, 0.957216608346604294854305150230, 1.37005452738535426922838548994, 2.27518197208888701136112060493, 2.43506758899085231630649874501, 3.19507717556796983472041937912, 3.66066401784555330183746257264, 4.07542643680189639343869062064, 4.29256231449631530588458288255, 4.94550006468922623269185147684, 5.08416728227276067570101076998, 5.50409982882993187220297422579, 5.91226467228514597381191208541, 6.08890248327757243666290427977, 6.54785454165669105975724525716, 7.05417338786111630305243556601, 7.36133899707079316986979821576, 7.64893602029280037813170848950, 7.74376961124057779346326971116, 8.601210787736251452283567114581

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