Properties

Label 4-95e4-1.1-c1e2-0-9
Degree $4$
Conductor $81450625$
Sign $1$
Analytic cond. $5193.36$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 4·7-s − 6·9-s − 3·16-s + 4·17-s − 12·23-s + 4·28-s − 6·36-s + 12·43-s + 4·47-s − 2·49-s − 20·61-s − 24·63-s − 7·64-s + 4·68-s − 28·73-s + 27·81-s − 28·83-s − 12·92-s − 20·101-s − 12·112-s + 16·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.51·7-s − 2·9-s − 3/4·16-s + 0.970·17-s − 2.50·23-s + 0.755·28-s − 36-s + 1.82·43-s + 0.583·47-s − 2/7·49-s − 2.56·61-s − 3.02·63-s − 7/8·64-s + 0.485·68-s − 3.27·73-s + 3·81-s − 3.07·83-s − 1.25·92-s − 1.99·101-s − 1.13·112-s + 1.46·119-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81450625\)    =    \(5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(5193.36\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 81450625,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad5 \( 1 \)
19 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
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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

−7.81208363530236726404997665037, −7.38116270924417067237467240847, −6.93025724488558533499758945176, −6.39921927696296215619010693838, −6.09651207259395221177883347994, −5.75147952375285553469512159742, −5.65872611871033374739293143102, −5.33126707827038524156288453971, −4.59405259041441511593151909866, −4.57098776577191378324447689144, −4.08284126326574961570411921192, −3.71180799570969580182241767788, −3.03820340865098716968356653887, −2.73742954127320276525294729056, −2.58196722155030030635340973388, −1.84623628770838907716813073375, −1.66595214228068149650120013126, −1.12436510392985989269873786035, 0, 0, 1.12436510392985989269873786035, 1.66595214228068149650120013126, 1.84623628770838907716813073375, 2.58196722155030030635340973388, 2.73742954127320276525294729056, 3.03820340865098716968356653887, 3.71180799570969580182241767788, 4.08284126326574961570411921192, 4.57098776577191378324447689144, 4.59405259041441511593151909866, 5.33126707827038524156288453971, 5.65872611871033374739293143102, 5.75147952375285553469512159742, 6.09651207259395221177883347994, 6.39921927696296215619010693838, 6.93025724488558533499758945176, 7.38116270924417067237467240847, 7.81208363530236726404997665037

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