Properties

Label 4-5070e2-1.1-c1e2-0-2
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 + 4·17-s − 16·23-s − 25-s + 4·27-s + 4·29-s − 3·36-s − 24·43-s + 2·48-s − 2·49-s + 8·51-s + 20·53-s − 20·61-s − 64-s − 4·68-s − 32·69-s − 2·75-s − 16·79-s + 5·81-s + 8·87-s + 16·92-s + 100-s − 20·101-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s + 0.970·17-s − 3.33·23-s − 1/5·25-s + 0.769·27-s + 0.742·29-s − 1/2·36-s − 3.65·43-s + 0.288·48-s − 2/7·49-s + 1.12·51-s + 2.74·53-s − 2.56·61-s − 1/8·64-s − 0.485·68-s − 3.85·69-s − 0.230·75-s − 1.80·79-s + 5/9·81-s + 0.857·87-s + 1.66·92-s + 1/10·100-s − 1.99·101-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

\(L(1)\) \(\approx\) \(1.525864898\)
\(L(\frac12)\) \(\approx\) \(1.525864898\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−8.454568999041999285548090878256, −8.156534110156454528291463361006, −7.902779101384107378741333793771, −7.38974578229457086264478775725, −7.16404990992205796268208567653, −6.55477693633447694789149361466, −6.36681463879856556672251198774, −5.81744961166176221543510098640, −5.50535974498790619140837492969, −5.16922959109749358835537838779, −4.46948761712770069045514210303, −4.31199544402349728218557219639, −3.93742026584893608206283595605, −3.41461401347423865146825602999, −3.23796013966328258236053777514, −2.64268412581824663025404423360, −2.16965592687440240716886965186, −1.61429504050987896324182304044, −1.37644351106380768032526982252, −0.29520426968162943152674472081, 0.29520426968162943152674472081, 1.37644351106380768032526982252, 1.61429504050987896324182304044, 2.16965592687440240716886965186, 2.64268412581824663025404423360, 3.23796013966328258236053777514, 3.41461401347423865146825602999, 3.93742026584893608206283595605, 4.31199544402349728218557219639, 4.46948761712770069045514210303, 5.16922959109749358835537838779, 5.50535974498790619140837492969, 5.81744961166176221543510098640, 6.36681463879856556672251198774, 6.55477693633447694789149361466, 7.16404990992205796268208567653, 7.38974578229457086264478775725, 7.902779101384107378741333793771, 8.156534110156454528291463361006, 8.454568999041999285548090878256

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