Properties

Label 6-160e3-1.1-c5e3-0-0
Degree $6$
Conductor $4096000$
Sign $1$
Analytic cond. $16898.2$
Root an. cond. $5.06570$
Motivic weight $5$
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  + 10·3-s − 75·5-s − 6·7-s − 81·9-s − 396·11-s − 354·13-s − 750·15-s + 1.15e3·17-s + 3.19e3·19-s − 60·21-s + 6.12e3·23-s + 3.75e3·25-s + 2.64e3·27-s + 426·29-s + 3.27e3·31-s − 3.96e3·33-s + 450·35-s − 1.15e4·37-s − 3.54e3·39-s + 1.24e4·41-s + 2.63e4·43-s + 6.07e3·45-s + 3.67e4·47-s − 2.71e4·49-s + 1.15e4·51-s − 2.11e4·53-s + 2.97e4·55-s + ⋯
L(s)  = 1  + 0.641·3-s − 1.34·5-s − 0.0462·7-s − 1/3·9-s − 0.986·11-s − 0.580·13-s − 0.860·15-s + 0.971·17-s + 2.02·19-s − 0.0296·21-s + 2.41·23-s + 6/5·25-s + 0.699·27-s + 0.0940·29-s + 0.612·31-s − 0.633·33-s + 0.0620·35-s − 1.38·37-s − 0.372·39-s + 1.15·41-s + 2.17·43-s + 0.447·45-s + 2.42·47-s − 1.61·49-s + 0.623·51-s − 1.03·53-s + 1.32·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(4096000\)    =    \(2^{15} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(16898.2\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 4096000,\ (\ :5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.249046887\)
\(L(\frac12)\) \(\approx\) \(4.249046887\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p^{2} T )^{3} \)
good3$S_4\times C_2$ \( 1 - 10 T + 181 T^{2} - 1756 p T^{3} + 181 p^{5} T^{4} - 10 p^{10} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 + 6 T + 3879 p T^{2} + 1137532 T^{3} + 3879 p^{6} T^{4} + 6 p^{10} T^{5} + p^{15} T^{6} \)
11$S_4\times C_2$ \( 1 + 36 p T + 327873 T^{2} + 67617992 T^{3} + 327873 p^{5} T^{4} + 36 p^{11} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 + 354 T + 845379 T^{2} + 291738444 T^{3} + 845379 p^{5} T^{4} + 354 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 - 1158 T + 1914111 T^{2} - 544550612 T^{3} + 1914111 p^{5} T^{4} - 1158 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 - 168 p T + 9330057 T^{2} - 15933739216 T^{3} + 9330057 p^{5} T^{4} - 168 p^{11} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 - 6126 T + 29722593 T^{2} - 84141222540 T^{3} + 29722593 p^{5} T^{4} - 6126 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 - 426 T - 939693 T^{2} + 141773503364 T^{3} - 939693 p^{5} T^{4} - 426 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 - 3276 T + 2292813 T^{2} + 41639420248 T^{3} + 2292813 p^{5} T^{4} - 3276 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 + 11562 T + 90623691 T^{2} + 525372493468 T^{3} + 90623691 p^{5} T^{4} + 11562 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 384843783 p^{5} T^{4} - 12450 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 26346 T + 640605069 T^{2} - 8098987233524 T^{3} + 640605069 p^{5} T^{4} - 26346 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 - 36762 T + 1119958377 T^{2} - 18490559326820 T^{3} + 1119958377 p^{5} T^{4} - 36762 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 + 21162 T + 395789499 T^{2} - 881498066468 T^{3} + 395789499 p^{5} T^{4} + 21162 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 - 35040 T + 1757220897 T^{2} - 50989349593920 T^{3} + 1757220897 p^{5} T^{4} - 35040 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 393086643 p^{5} T^{4} + 24138 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 + 9570 T + 659335509 T^{2} + 83080838420484 T^{3} + 659335509 p^{5} T^{4} + 9570 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 88092 T + 7289446053 T^{2} - 329541325840584 T^{3} + 7289446053 p^{5} T^{4} - 88092 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 - 66750 T + 3875077479 T^{2} - 111360074258500 T^{3} + 3875077479 p^{5} T^{4} - 66750 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 + 92952 T + 11164448877 T^{2} + 573846024396496 T^{3} + 11164448877 p^{5} T^{4} + 92952 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 - 30258 T + 4293702405 T^{2} - 426012342532708 T^{3} + 4293702405 p^{5} T^{4} - 30258 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} + 26445328791 p^{5} T^{4} - 172686 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 - 170910 T + 30283966671 T^{2} - 2852314667192740 T^{3} + 30283966671 p^{5} T^{4} - 170910 p^{10} T^{5} + p^{15} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.77595980459570363843061478455, −10.40863937391579484925089038376, −9.791577168237250152630951500107, −9.704948015342368619048192252063, −9.050740148067776304426882051682, −8.814857408560453307414584951366, −8.792722971364665246147493874558, −7.81977183620813337860427884549, −7.79860942012218930363787089575, −7.67541402692812987212909204492, −7.32489583251520492713429655407, −6.64819563871903017779396278895, −6.56074574522555306228522901662, −5.65216840625499503997222915924, −5.19288498165303801902364402961, −5.14264646244187625709568382412, −4.70792690017935232102777598009, −3.88593577394161215316757927966, −3.69003441553887275972559230985, −2.91831087837439195192114925289, −2.87925918490213238237340748663, −2.54417857700866150955301210052, −1.36419258460414701214751002118, −0.74244313692243571320409817133, −0.60798069102817241945210999279, 0.60798069102817241945210999279, 0.74244313692243571320409817133, 1.36419258460414701214751002118, 2.54417857700866150955301210052, 2.87925918490213238237340748663, 2.91831087837439195192114925289, 3.69003441553887275972559230985, 3.88593577394161215316757927966, 4.70792690017935232102777598009, 5.14264646244187625709568382412, 5.19288498165303801902364402961, 5.65216840625499503997222915924, 6.56074574522555306228522901662, 6.64819563871903017779396278895, 7.32489583251520492713429655407, 7.67541402692812987212909204492, 7.79860942012218930363787089575, 7.81977183620813337860427884549, 8.792722971364665246147493874558, 8.814857408560453307414584951366, 9.050740148067776304426882051682, 9.704948015342368619048192252063, 9.791577168237250152630951500107, 10.40863937391579484925089038376, 10.77595980459570363843061478455

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