Properties

Label 4-15e4-1.1-c11e2-0-0
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $29886.5$
Root an. cond. $13.1482$
Motivic weight $11$
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  + 3.52e3·4-s − 1.06e6·11-s + 8.19e6·16-s − 2.13e7·19-s + 2.56e8·29-s − 1.05e8·31-s − 6.16e8·41-s − 3.76e9·44-s + 3.67e9·49-s − 1.03e10·59-s + 1.39e10·61-s + 1.40e10·64-s − 1.95e10·71-s − 7.50e10·76-s − 7.62e10·79-s − 4.99e10·89-s − 1.63e11·101-s − 1.46e11·109-s + 9.03e11·116-s + 2.86e11·121-s − 3.72e11·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.71·4-s − 2.00·11-s + 1.95·16-s − 1.97·19-s + 2.32·29-s − 0.663·31-s − 0.830·41-s − 3.44·44-s + 1.85·49-s − 1.88·59-s + 2.10·61-s + 1.63·64-s − 1.28·71-s − 3.39·76-s − 2.78·79-s − 0.948·89-s − 1.54·101-s − 0.914·109-s + 3.99·116-s + 1.00·121-s − 1.13·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(29886.5\)
Root analytic conductor: \(13.1482\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50625,\ (\ :11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.004873614963\)
\(L(\frac12)\) \(\approx\) \(0.004873614963\)
\(L(\frac{13}{2})\) 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$C_2^2$ \( 1 - 55 p^{6} T^{2} + p^{22} T^{4} \)
7$C_2^2$ \( 1 - 74985550 p^{2} T^{2} + p^{22} T^{4} \)
11$C_2$ \( ( 1 + 534612 T + p^{11} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3250539591430 T^{2} + p^{22} T^{4} \)
17$C_2^2$ \( 1 - 20851868202910 T^{2} + p^{22} T^{4} \)
19$C_2$ \( ( 1 + 10661420 T + p^{11} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 1558047924961870 T^{2} + p^{22} T^{4} \)
29$C_2$ \( ( 1 - 128406630 T + p^{11} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 52843168 T + p^{11} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 322633551760058230 T^{2} + p^{22} T^{4} \)
41$C_2$ \( ( 1 + 308120442 T + p^{11} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 1858294189067944150 T^{2} + p^{22} T^{4} \)
47$C_2^2$ \( 1 + 2277523508785437410 T^{2} + p^{22} T^{4} \)
53$C_2^2$ \( 1 - 15990678067626115990 T^{2} + p^{22} T^{4} \)
59$C_2$ \( ( 1 + 5189203740 T + p^{11} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6956478662 T + p^{11} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 4573302143790884710 T^{2} + p^{22} T^{4} \)
71$C_2$ \( ( 1 + 9791485272 T + p^{11} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(62\!\cdots\!70\)\( T^{2} + p^{22} T^{4} \)
79$C_2$ \( ( 1 + 38116845680 T + p^{11} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(17\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \)
89$C_2$ \( ( 1 + 24992917110 T + p^{11} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(86\!\cdots\!90\)\( T^{2} + p^{22} T^{4} \)
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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

−10.61792105709686980182465100156, −10.24616687341048772272428663671, −9.936283860459419913211570258474, −8.683487524329814754665786790131, −8.674689439542522832357770919741, −7.912691193284548743896185801747, −7.63529369793335355681269616128, −6.96480550976476341252107537567, −6.65639521650024516712248753959, −6.11389705859075046504692260448, −5.60406201079782991377381774993, −5.09817895199712814539951895883, −4.42261216643305898851799266767, −3.80397859402501414126561527450, −2.90450685462546244542919448052, −2.50092343032972517655854274289, −2.47825163432391381323086690244, −1.58088423171914209355537115453, −1.12808692560075699559936570275, −0.01188972895001192993101294083, 0.01188972895001192993101294083, 1.12808692560075699559936570275, 1.58088423171914209355537115453, 2.47825163432391381323086690244, 2.50092343032972517655854274289, 2.90450685462546244542919448052, 3.80397859402501414126561527450, 4.42261216643305898851799266767, 5.09817895199712814539951895883, 5.60406201079782991377381774993, 6.11389705859075046504692260448, 6.65639521650024516712248753959, 6.96480550976476341252107537567, 7.63529369793335355681269616128, 7.912691193284548743896185801747, 8.674689439542522832357770919741, 8.683487524329814754665786790131, 9.936283860459419913211570258474, 10.24616687341048772272428663671, 10.61792105709686980182465100156

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