Properties

Label 4-5070e2-1.1-c1e2-0-17
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 $2$

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 − 16·17-s − 12·23-s − 25-s − 4·27-s − 8·29-s − 3·36-s + 8·43-s − 2·48-s + 10·49-s + 32·51-s − 20·53-s − 20·61-s − 64-s + 16·68-s + 24·69-s + 2·75-s + 16·79-s + 5·81-s + 16·87-s + 12·92-s + 100-s + 32·101-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s − 3.88·17-s − 2.50·23-s − 1/5·25-s − 0.769·27-s − 1.48·29-s − 1/2·36-s + 1.21·43-s − 0.288·48-s + 10/7·49-s + 4.48·51-s − 2.74·53-s − 2.56·61-s − 1/8·64-s + 1.94·68-s + 2.88·69-s + 0.230·75-s + 1.80·79-s + 5/9·81-s + 1.71·87-s + 1.25·92-s + 1/10·100-s + 3.18·101-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: \(2\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−7.890723838570164006515342968873, −7.69524363933831705678297303833, −7.28114190331287182662159880881, −7.00385176977187641524415900809, −6.33117236701521112396022104275, −6.23406995952728775956924325643, −5.91826796095453046104039957369, −5.79640421327515474280345611260, −4.82102169894939032429475651892, −4.77144264386310586897385757542, −4.35644715598951209367070405300, −4.30093706890926341643824962610, −3.51802980583544664536812234277, −3.36633959734645076296051979797, −2.19284819031003011480493001001, −2.08811721803671611353124900959, −1.90945856466375441765065330650, −0.803750271251906704292505362934, 0, 0, 0.803750271251906704292505362934, 1.90945856466375441765065330650, 2.08811721803671611353124900959, 2.19284819031003011480493001001, 3.36633959734645076296051979797, 3.51802980583544664536812234277, 4.30093706890926341643824962610, 4.35644715598951209367070405300, 4.77144264386310586897385757542, 4.82102169894939032429475651892, 5.79640421327515474280345611260, 5.91826796095453046104039957369, 6.23406995952728775956924325643, 6.33117236701521112396022104275, 7.00385176977187641524415900809, 7.28114190331287182662159880881, 7.69524363933831705678297303833, 7.890723838570164006515342968873

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