Properties

Label 8-14e4-1.1-c8e4-0-0
Degree $8$
Conductor $38416$
Sign $1$
Analytic cond. $1058.04$
Root an. cond. $2.38815$
Motivic weight $8$
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  + 256·4-s − 6.07e3·7-s + 6.27e3·9-s − 1.35e4·11-s + 4.91e4·16-s − 8.94e5·23-s + 1.38e6·25-s − 1.55e6·28-s + 3.17e5·29-s + 1.60e6·36-s − 2.49e6·37-s + 9.18e6·43-s − 3.47e6·44-s + 1.89e7·49-s − 3.87e7·53-s − 3.81e7·63-s + 8.38e6·64-s − 1.23e7·67-s + 6.21e7·71-s + 8.23e7·77-s + 2.48e7·79-s − 2.04e7·81-s − 2.28e8·92-s − 8.51e7·99-s + 3.55e8·100-s − 5.10e8·107-s + 1.36e8·109-s + ⋯
L(s)  = 1  + 4-s − 2.53·7-s + 0.956·9-s − 0.926·11-s + 3/4·16-s − 3.19·23-s + 3.55·25-s − 2.53·28-s + 0.448·29-s + 0.956·36-s − 1.33·37-s + 2.68·43-s − 0.926·44-s + 3.28·49-s − 4.90·53-s − 2.42·63-s + 1/2·64-s − 0.611·67-s + 2.44·71-s + 2.34·77-s + 0.639·79-s − 0.475·81-s − 3.19·92-s − 0.885·99-s + 3.55·100-s − 3.89·107-s + 0.964·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(38416\)    =    \(2^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1058.04\)
Root analytic conductor: \(2.38815\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 38416,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.138763152\)
\(L(\frac12)\) \(\approx\) \(1.138763152\)
\(L(5)\) 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$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
7$D_{4}$ \( 1 + 124 p^{2} T + 7494 p^{4} T^{2} + 124 p^{10} T^{3} + p^{16} T^{4} \)
good3$C_2^2 \wr C_2$ \( 1 - 2092 p T^{2} + 6649942 p^{2} T^{4} - 2092 p^{17} T^{6} + p^{32} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 55588 p^{2} T^{2} + 783643216902 T^{4} - 55588 p^{18} T^{6} + p^{32} T^{8} \)
11$D_{4}$ \( ( 1 + 6780 T + 385462214 T^{2} + 6780 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
13$C_2^2 \wr C_2$ \( 1 - 1612887940 T^{2} + 1300734062657673990 T^{4} - 1612887940 p^{16} T^{6} + p^{32} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 16023366148 T^{2} + \)\(14\!\cdots\!66\)\( T^{4} - 16023366148 p^{16} T^{6} + p^{32} T^{8} \)
19$C_2^2 \wr C_2$ \( 1 + 26834594684 T^{2} + \)\(74\!\cdots\!38\)\( T^{4} + 26834594684 p^{16} T^{6} + p^{32} T^{8} \)
23$D_{4}$ \( ( 1 + 447036 T + 202011057158 T^{2} + 447036 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 158532 T + 420149466566 T^{2} - 158532 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 - 1257474960388 T^{2} + \)\(17\!\cdots\!46\)\( T^{4} - 1257474960388 p^{16} T^{6} + p^{32} T^{8} \)
37$D_{4}$ \( ( 1 + 1247548 T + 7156715791686 T^{2} + 1247548 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 12045003373828 T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - 12045003373828 p^{16} T^{6} + p^{32} T^{8} \)
43$D_{4}$ \( ( 1 - 4593284 T + 25927747162566 T^{2} - 4593284 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - 29473105285636 T^{2} + \)\(52\!\cdots\!38\)\( T^{4} - 29473105285636 p^{16} T^{6} + p^{32} T^{8} \)
53$D_{4}$ \( ( 1 + 19363644 T + 215363679361094 T^{2} + 19363644 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 - 468430280695684 T^{2} + \)\(97\!\cdots\!46\)\( T^{4} - 468430280695684 p^{16} T^{6} + p^{32} T^{8} \)
61$C_2^2 \wr C_2$ \( 1 - 306906143267716 T^{2} + \)\(82\!\cdots\!18\)\( T^{4} - 306906143267716 p^{16} T^{6} + p^{32} T^{8} \)
67$D_{4}$ \( ( 1 + 6160124 T - 450645761195706 T^{2} + 6160124 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 31084356 T + 692364955411334 T^{2} - 31084356 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 - 1018814792244484 T^{2} + \)\(36\!\cdots\!18\)\( T^{4} - 1018814792244484 p^{16} T^{6} + p^{32} T^{8} \)
79$D_{4}$ \( ( 1 - 12444868 T + 2983388778012678 T^{2} - 12444868 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - 2088432522150532 T^{2} + \)\(90\!\cdots\!50\)\( T^{4} - 2088432522150532 p^{16} T^{6} + p^{32} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - 6583269594928900 T^{2} + \)\(30\!\cdots\!10\)\( T^{4} - 6583269594928900 p^{16} T^{6} + p^{32} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 - 14985098969373700 T^{2} + \)\(11\!\cdots\!70\)\( T^{4} - 14985098969373700 p^{16} T^{6} + p^{32} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.72956230224811052319299643123, −12.61452822494309223187078393100, −12.31862639609115919139896814344, −12.24855051228745930910965168794, −11.29028482681189161730413531218, −10.86269329183905322080813789172, −10.52757074167974710091295951007, −10.27880113191616934778718022472, −9.883237269673085455580043929028, −9.292000889738062764498072963176, −9.280201888079765633283944613021, −8.322945381620476980189064903073, −7.893278058301279733998532062666, −7.45054779177891356455027304432, −6.78542561783607115665751063069, −6.49819014705709448408703529653, −6.33203046282976399299985453299, −5.63477003052124937149897570241, −4.87248896752027032418138615843, −4.10577422569934297697525647882, −3.37856766161271038168758780583, −2.91607987279349019406377410548, −2.36827426394006627205473115982, −1.35052716356263521759968408162, −0.32048335661327030632103656393, 0.32048335661327030632103656393, 1.35052716356263521759968408162, 2.36827426394006627205473115982, 2.91607987279349019406377410548, 3.37856766161271038168758780583, 4.10577422569934297697525647882, 4.87248896752027032418138615843, 5.63477003052124937149897570241, 6.33203046282976399299985453299, 6.49819014705709448408703529653, 6.78542561783607115665751063069, 7.45054779177891356455027304432, 7.893278058301279733998532062666, 8.322945381620476980189064903073, 9.280201888079765633283944613021, 9.292000889738062764498072963176, 9.883237269673085455580043929028, 10.27880113191616934778718022472, 10.52757074167974710091295951007, 10.86269329183905322080813789172, 11.29028482681189161730413531218, 12.24855051228745930910965168794, 12.31862639609115919139896814344, 12.61452822494309223187078393100, 12.72956230224811052319299643123

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