Properties

Label 4-5070e2-1.1-c1e2-0-4
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s − 12·17-s − 6·23-s − 25-s + 4·27-s + 6·29-s − 3·36-s + 2·43-s + 2·48-s + 10·49-s − 24·51-s − 12·53-s + 4·61-s − 64-s + 12·68-s − 12·69-s − 2·75-s + 10·79-s + 5·81-s + 12·87-s + 6·92-s + 100-s − 12·101-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s − 2.91·17-s − 1.25·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 1/2·36-s + 0.304·43-s + 0.288·48-s + 10/7·49-s − 3.36·51-s − 1.64·53-s + 0.512·61-s − 1/8·64-s + 1.45·68-s − 1.44·69-s − 0.230·75-s + 1.12·79-s + 5/9·81-s + 1.28·87-s + 0.625·92-s + 1/10·100-s − 1.19·101-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: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.759956498\)
\(L(\frac12)\) \(\approx\) \(1.759956498\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.349542012464486739397088149720, −8.326249008305314558683950913306, −7.79554238499848118647564076581, −7.39853799603499075590438023952, −6.92915391690841619223447171287, −6.76488647011071174747610385004, −6.25230391679810651195193028337, −6.05740187538049644759979417788, −5.44939507567546335458426373303, −4.95852894655401897808059470391, −4.47503395124674126951283021363, −4.42947053621484210932482280760, −3.84959798874664381953104334032, −3.72825288040742446441310192257, −2.86257019471197360452941215465, −2.73372979041411725114475866718, −2.04866679336458395753259929269, −1.97452111022205197322206413750, −1.15658001409719249031054745265, −0.33373083231994877022887768936, 0.33373083231994877022887768936, 1.15658001409719249031054745265, 1.97452111022205197322206413750, 2.04866679336458395753259929269, 2.73372979041411725114475866718, 2.86257019471197360452941215465, 3.72825288040742446441310192257, 3.84959798874664381953104334032, 4.42947053621484210932482280760, 4.47503395124674126951283021363, 4.95852894655401897808059470391, 5.44939507567546335458426373303, 6.05740187538049644759979417788, 6.25230391679810651195193028337, 6.76488647011071174747610385004, 6.92915391690841619223447171287, 7.39853799603499075590438023952, 7.79554238499848118647564076581, 8.326249008305314558683950913306, 8.349542012464486739397088149720

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