Properties

Label 4-14e4-1.1-c3e2-0-8
Degree $4$
Conductor $38416$
Sign $1$
Analytic cond. $133.734$
Root an. cond. $3.40064$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 52·9-s − 52·11-s − 296·23-s − 242·25-s − 236·29-s − 508·37-s + 244·43-s − 340·53-s + 840·67-s + 840·71-s + 2.10e3·79-s + 1.97e3·81-s + 2.70e3·99-s + 1.76e3·107-s + 1.19e3·109-s − 3.09e3·113-s − 634·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.24e3·169-s + ⋯
L(s)  = 1  − 1.92·9-s − 1.42·11-s − 2.68·23-s − 1.93·25-s − 1.51·29-s − 2.25·37-s + 0.865·43-s − 0.881·53-s + 1.53·67-s + 1.40·71-s + 2.99·79-s + 2.70·81-s + 2.74·99-s + 1.59·107-s + 1.05·109-s − 2.57·113-s − 0.476·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.47·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(38416\)    =    \(2^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(133.734\)
Root analytic conductor: \(3.40064\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 38416,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 52 T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 242 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 26 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 3242 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 832 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 4740 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 148 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 118 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 28618 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 254 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 129392 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 122 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 112598 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 170 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 318308 T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 84162 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 420 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 420 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 116784 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1052 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 957676 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 358688 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 1725888 T^{2} + p^{6} T^{4} \)
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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

−11.71712420609773471690665250998, −11.39060317175056551334154486048, −10.79267688741083356672135526705, −10.47487367177661302017475641176, −9.655876193632681288055235369055, −9.528622967899325740229695544400, −8.577149791689350606159057502039, −8.277507422016815221742895073099, −7.77066322685964034313537065424, −7.46661262613619143887952057991, −6.22022877222351440071548759067, −6.09514041788396441401649172021, −5.31995382662879687573706734978, −5.13323057035691803607918386971, −3.71782982636151383795871771327, −3.62332699959918104552027039313, −2.29525485551279358489498593280, −2.14463166975265245849982783769, 0, 0, 2.14463166975265245849982783769, 2.29525485551279358489498593280, 3.62332699959918104552027039313, 3.71782982636151383795871771327, 5.13323057035691803607918386971, 5.31995382662879687573706734978, 6.09514041788396441401649172021, 6.22022877222351440071548759067, 7.46661262613619143887952057991, 7.77066322685964034313537065424, 8.277507422016815221742895073099, 8.577149791689350606159057502039, 9.528622967899325740229695544400, 9.655876193632681288055235369055, 10.47487367177661302017475641176, 10.79267688741083356672135526705, 11.39060317175056551334154486048, 11.71712420609773471690665250998

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