Properties

Label 4-570e2-1.1-c3e2-0-0
Degree $4$
Conductor $324900$
Sign $1$
Analytic cond. $1131.05$
Root an. cond. $5.79923$
Motivic weight $3$
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  + 2·2-s + 3·3-s + 5·5-s + 6·6-s − 38·7-s − 8·8-s + 10·10-s + 30·11-s − 50·13-s − 76·14-s + 15·15-s − 16·16-s + 42·17-s − 133·19-s − 114·21-s + 60·22-s + 45·23-s − 24·24-s − 100·26-s − 27·27-s + 108·29-s + 30·30-s − 392·31-s + 90·33-s + 84·34-s − 190·35-s − 86·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s − 2.05·7-s − 0.353·8-s + 0.316·10-s + 0.822·11-s − 1.06·13-s − 1.45·14-s + 0.258·15-s − 1/4·16-s + 0.599·17-s − 1.60·19-s − 1.18·21-s + 0.581·22-s + 0.407·23-s − 0.204·24-s − 0.754·26-s − 0.192·27-s + 0.691·29-s + 0.182·30-s − 2.27·31-s + 0.474·33-s + 0.423·34-s − 0.917·35-s − 0.382·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1131.05\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324900,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3031829519\)
\(L(\frac12)\) \(\approx\) \(0.3031829519\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
3$C_2$ \( 1 - p T + p^{2} T^{2} \)
5$C_2$ \( 1 - p T + p^{2} T^{2} \)
19$C_2$ \( 1 + 7 p T + p^{3} T^{2} \)
good7$C_2$ \( ( 1 + 19 T + p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 15 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 50 T + 303 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 42 T - 3149 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 45 T - 10142 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 108 T - 12725 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 196 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 43 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 213 T - 23552 T^{2} + 213 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 338 T + 34737 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 240 T - 46223 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 453 T + 56332 T^{2} - 453 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 180 T - 172979 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 98 T - 217377 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 110 T - 288663 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 + 204 T - 316295 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \)
73$C_2^2$ \( 1 - 760 T + 188583 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 214 T - 447243 T^{2} - 214 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 198 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 615 T - 326744 T^{2} + 615 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 52 T - 909969 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.80562285169270948462841633812, −9.883072383970417231670499339873, −9.753018396200894396769828478234, −9.205464819313940605246380569806, −9.047288885128265762125131120285, −8.380711283718082592470660524679, −8.008242755199386979338620774185, −7.04028121828366431354255624943, −6.88972952559080363792953966084, −6.56611998421207075808671927694, −5.98538806538154602295588118269, −5.49379007846642715055079618938, −5.00340450624673127185850464065, −4.29926175925290325435219896903, −3.63582065351854345019735097249, −3.47055270360893680639543755616, −2.80382723753890255913694175730, −2.26374046104369393761634253107, −1.48326773947469385932243162689, −0.13285472666336875541499696332, 0.13285472666336875541499696332, 1.48326773947469385932243162689, 2.26374046104369393761634253107, 2.80382723753890255913694175730, 3.47055270360893680639543755616, 3.63582065351854345019735097249, 4.29926175925290325435219896903, 5.00340450624673127185850464065, 5.49379007846642715055079618938, 5.98538806538154602295588118269, 6.56611998421207075808671927694, 6.88972952559080363792953966084, 7.04028121828366431354255624943, 8.008242755199386979338620774185, 8.380711283718082592470660524679, 9.047288885128265762125131120285, 9.205464819313940605246380569806, 9.753018396200894396769828478234, 9.883072383970417231670499339873, 10.80562285169270948462841633812

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