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Properties

Label 4-39e2-1.1-c14e2-0-0
Degree $4$
Conductor $1521$
Sign $1$
Analytic cond. $2351.11$
Root an. cond. $6.96335$
Motivic weight $14$
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  + 4.37e3·3-s − 3.27e4·4-s + 1.43e7·9-s − 1.43e8·12-s − 1.25e8·13-s + 8.05e8·16-s + 3.37e9·25-s + 4.18e10·27-s − 4.70e11·36-s − 5.48e11·39-s + 3.96e11·43-s + 3.52e12·48-s + 1.35e12·49-s + 4.11e12·52-s + 1.14e13·61-s − 1.75e13·64-s + 1.47e13·75-s + 5.95e13·79-s + 1.14e14·81-s − 1.10e14·100-s − 4.86e14·103-s − 1.37e15·108-s − 1.80e15·117-s − 2.81e13·121-s + 127-s + 1.73e15·129-s + 131-s + ⋯
L(s)  = 1  + 2·3-s − 1.99·4-s + 3·9-s − 3.99·12-s − 2·13-s + 2.99·16-s + 0.552·25-s + 4·27-s − 5.99·36-s − 4·39-s + 1.45·43-s + 5.99·48-s + 2·49-s + 3.99·52-s + 3.62·61-s − 3.99·64-s + 1.10·75-s + 3.10·79-s + 5·81-s − 1.10·100-s − 3.95·103-s − 7.99·108-s − 6·117-s − 0.0741·121-s + 2.91·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2351.11\)
Root analytic conductor: \(6.96335\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1521,\ (\ :7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(3.782418443\)
\(L(\frac12)\) \(\approx\) \(3.782418443\)
\(L(8)\) 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$C_1$ \( ( 1 - p^{7} T )^{2} \)
13$C_1$ \( ( 1 + p^{7} T )^{2} \)
good2$C_2^2$ \( 1 + 32765 T^{2} + p^{28} T^{4} \)
5$C_2^2$ \( 1 - 3374358622 T^{2} + p^{28} T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
11$C_2^2$ \( 1 + 28162265432930 T^{2} + p^{28} T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
41$C_2^2$ \( 1 - \)\(68\!\cdots\!10\)\( T^{2} + p^{28} T^{4} \)
43$C_2$ \( ( 1 - 198255592010 T + p^{14} T^{2} )^{2} \)
47$C_2^2$ \( 1 + \)\(26\!\cdots\!10\)\( T^{2} + p^{28} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
59$C_2^2$ \( 1 - \)\(81\!\cdots\!30\)\( T^{2} + p^{28} T^{4} \)
61$C_2$ \( ( 1 - 5700982145110 T + p^{14} T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
71$C_2^2$ \( 1 + \)\(71\!\cdots\!30\)\( T^{2} + p^{28} T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
79$C_2$ \( ( 1 - 29788905182350 T + p^{14} T^{2} )^{2} \)
83$C_2^2$ \( 1 + \)\(12\!\cdots\!90\)\( T^{2} + p^{28} T^{4} \)
89$C_2^2$ \( 1 - \)\(38\!\cdots\!50\)\( T^{2} + p^{28} T^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
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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

−13.49525926762746569112352439673, −13.11825932957827737166485729377, −12.38879205497682236807278171660, −12.27597927001332299393894525347, −10.55515767247987531254507212158, −10.06847652368973444463180693078, −9.469985956337611801980297760896, −9.251885958270283468749019585911, −8.603237048211633459432375177569, −8.026458239756005692040775357477, −7.54309301914445727989061444222, −6.84829578099889709156071473892, −5.29597275633829177323260752017, −4.90744289252111687769622786536, −3.94249874201541902170757188776, −3.91193402898343632766236241748, −2.73147841916045218515278534318, −2.33505775411725703903376467415, −1.13790284804747635805943028020, −0.54029512947454796429196264000, 0.54029512947454796429196264000, 1.13790284804747635805943028020, 2.33505775411725703903376467415, 2.73147841916045218515278534318, 3.91193402898343632766236241748, 3.94249874201541902170757188776, 4.90744289252111687769622786536, 5.29597275633829177323260752017, 6.84829578099889709156071473892, 7.54309301914445727989061444222, 8.026458239756005692040775357477, 8.603237048211633459432375177569, 9.251885958270283468749019585911, 9.469985956337611801980297760896, 10.06847652368973444463180693078, 10.55515767247987531254507212158, 12.27597927001332299393894525347, 12.38879205497682236807278171660, 13.11825932957827737166485729377, 13.49525926762746569112352439673

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