Properties

Label 4-5070e2-1.1-c1e2-0-11
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 + 6·23-s − 25-s − 4·27-s − 2·29-s − 3·36-s − 10·43-s − 2·48-s + 10·49-s + 8·51-s + 28·53-s − 20·61-s − 64-s + 4·68-s − 12·69-s + 2·75-s + 10·79-s + 5·81-s + 4·87-s − 6·92-s + 100-s − 28·101-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 + 1.25·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s − 1/2·36-s − 1.52·43-s − 0.288·48-s + 10/7·49-s + 1.12·51-s + 3.84·53-s − 2.56·61-s − 1/8·64-s + 0.485·68-s − 1.44·69-s + 0.230·75-s + 1.12·79-s + 5/9·81-s + 0.428·87-s − 0.625·92-s + 1/10·100-s − 2.78·101-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

\(L(1)\) \(\approx\) \(1.574265526\)
\(L(\frac12)\) \(\approx\) \(1.574265526\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.349473859479836628470175526276, −8.233431023005008434394608399032, −7.47771080534668747785694431991, −7.23737362341621947535403576022, −6.87456564460556029096669783655, −6.82400945157533984914835099993, −6.07817884334983330702626992726, −5.84042294540005821525179389638, −5.54895163414580302947710408022, −5.19341971990176875015362820515, −4.64005238413911631868106589778, −4.49918098312828972745521688840, −4.12672951268735207937340952666, −3.56059420124176434576397231008, −3.18601641569414944243713808471, −2.61956663383141666578452604900, −1.97122151110831402224764351116, −1.65360362690956341577402281819, −0.67003337186455975986198714174, −0.60179568925753632161282610233, 0.60179568925753632161282610233, 0.67003337186455975986198714174, 1.65360362690956341577402281819, 1.97122151110831402224764351116, 2.61956663383141666578452604900, 3.18601641569414944243713808471, 3.56059420124176434576397231008, 4.12672951268735207937340952666, 4.49918098312828972745521688840, 4.64005238413911631868106589778, 5.19341971990176875015362820515, 5.54895163414580302947710408022, 5.84042294540005821525179389638, 6.07817884334983330702626992726, 6.82400945157533984914835099993, 6.87456564460556029096669783655, 7.23737362341621947535403576022, 7.47771080534668747785694431991, 8.233431023005008434394608399032, 8.349473859479836628470175526276

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