Properties

Label 4-15e4-1.1-c3e2-0-4
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $176.237$
Root an. cond. $3.64354$
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  − 3·2-s + 4-s + 6·7-s − 3·8-s + 42·11-s + 78·13-s − 18·14-s + 9·16-s − 102·17-s + 56·19-s − 126·22-s + 48·23-s − 234·26-s + 6·28-s + 318·29-s + 52·31-s + 165·32-s + 306·34-s + 306·37-s − 168·38-s + 408·41-s + 120·43-s + 42·44-s − 144·46-s − 180·47-s − 290·49-s + 78·52-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s + 0.323·7-s − 0.132·8-s + 1.15·11-s + 1.66·13-s − 0.343·14-s + 9/64·16-s − 1.45·17-s + 0.676·19-s − 1.22·22-s + 0.435·23-s − 1.76·26-s + 0.0404·28-s + 2.03·29-s + 0.301·31-s + 0.911·32-s + 1.54·34-s + 1.35·37-s − 0.717·38-s + 1.55·41-s + 0.425·43-s + 0.143·44-s − 0.461·46-s − 0.558·47-s − 0.845·49-s + 0.208·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(176.237\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.542518300\)
\(L(\frac12)\) \(\approx\) \(1.542518300\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$D_{4}$ \( 1 + 3 T + p^{3} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 6 T + 326 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 42 T + 2734 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 6 p T + 5546 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 6 p T + 11402 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 56 T + 8598 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 48 T + 24254 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 318 T + 55978 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 52 T + 58782 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 306 T + 94826 T^{2} - 306 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 408 T + 177982 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 120 T + 68150 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 180 T + 128990 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 402 T + 269234 T^{2} + 402 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 732 T + 698582 T^{2} - 732 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 36 T + 384046 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1332 T + 1102034 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 380 T + 99678 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 984 T + 942182 T^{2} - 984 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1116 T + 1508758 T^{2} + 1116 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 768 T + 1382402 T^{2} + 768 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.80571952641259281737215735065, −11.35228670765698464519941519497, −11.09414326686794222870066265355, −10.69245416093424217944452775945, −9.826797481350842030637663743218, −9.507293083474311064757666705894, −9.103740165166302557944028315442, −8.686998973506634799237719726408, −8.069119485440187005198235074778, −8.032536892861511726770876066719, −6.84146947712091373497428141875, −6.47445085747211805538727891471, −6.23560279506738679801959823004, −5.27374910366502088213536785476, −4.47246280890572764860934712664, −4.07381497356330039798219886972, −3.18502310555794225382691145488, −2.34631098794546676435365039157, −1.16880540400715291829217987882, −0.76763121822473264237611057094, 0.76763121822473264237611057094, 1.16880540400715291829217987882, 2.34631098794546676435365039157, 3.18502310555794225382691145488, 4.07381497356330039798219886972, 4.47246280890572764860934712664, 5.27374910366502088213536785476, 6.23560279506738679801959823004, 6.47445085747211805538727891471, 6.84146947712091373497428141875, 8.032536892861511726770876066719, 8.069119485440187005198235074778, 8.686998973506634799237719726408, 9.103740165166302557944028315442, 9.507293083474311064757666705894, 9.826797481350842030637663743218, 10.69245416093424217944452775945, 11.09414326686794222870066265355, 11.35228670765698464519941519497, 11.80571952641259281737215735065

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