Properties

Degree 6
Conductor $ 17^{3} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4·3-s + 4-s − 8·5-s + 4·6-s + 22·7-s − 15·8-s − 3·9-s − 8·10-s − 28·11-s + 4·12-s + 30·13-s + 22·14-s − 32·15-s − 31·16-s − 51·17-s − 3·18-s + 80·19-s − 8·20-s + 88·21-s − 28·22-s + 142·23-s − 60·24-s − 267·25-s + 30·26-s − 76·27-s + 22·28-s + ⋯
L(s)  = 1  + 0.353·2-s + 0.769·3-s + 1/8·4-s − 0.715·5-s + 0.272·6-s + 1.18·7-s − 0.662·8-s − 1/9·9-s − 0.252·10-s − 0.767·11-s + 0.0962·12-s + 0.640·13-s + 0.419·14-s − 0.550·15-s − 0.484·16-s − 0.727·17-s − 0.0392·18-s + 0.965·19-s − 0.0894·20-s + 0.914·21-s − 0.271·22-s + 1.28·23-s − 0.510·24-s − 2.13·25-s + 0.226·26-s − 0.541·27-s + 0.148·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(4913\)    =    \(17^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{17} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 4913,\ (\ :3/2, 3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $1.40446$
$L(\frac12)$  $\approx$  $1.40446$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 17$, \(F_p\) is a polynomial of degree 6. If $p = 17$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad17$C_1$ \( ( 1 + p T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + p^{4} T^{3} - p^{6} T^{5} + p^{9} T^{6} \)
3$S_4\times C_2$ \( 1 - 4 T + 19 T^{2} - 4 p T^{3} + 19 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \)
5$S_4\times C_2$ \( 1 + 8 T + 331 T^{2} + 2032 T^{3} + 331 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 - 22 T + 891 T^{2} - 14300 T^{3} + 891 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 28 T + 2627 T^{2} + 69844 T^{3} + 2627 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 30 T + 5119 T^{2} - 141212 T^{3} + 5119 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 80 T + 15945 T^{2} - 757312 T^{3} + 15945 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 142 T + 20731 T^{2} - 1854884 T^{3} + 20731 p^{3} T^{4} - 142 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 456 T + 127075 T^{2} + 23761392 T^{3} + 127075 p^{3} T^{4} + 456 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 230 T + 77787 T^{2} - 13785468 T^{3} + 77787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 356 T + 133995 T^{2} - 29888184 T^{3} + 133995 p^{3} T^{4} - 356 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 294 T + 120199 T^{2} + 38886804 T^{3} + 120199 p^{3} T^{4} + 294 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 556 T + 289617 T^{2} - 81141512 T^{3} + 289617 p^{3} T^{4} - 556 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 640 T + 396797 T^{2} - 134564608 T^{3} + 396797 p^{3} T^{4} - 640 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 302 T + 293171 T^{2} - 71759636 T^{3} + 293171 p^{3} T^{4} - 302 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 636 T + 514369 T^{2} - 211823016 T^{3} + 514369 p^{3} T^{4} - 636 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 84 T + 556531 T^{2} + 31340024 T^{3} + 556531 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 1008 T + 967329 T^{2} - 607104160 T^{3} + 967329 p^{3} T^{4} - 1008 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 402 T + 483859 T^{2} + 12894428 T^{3} + 483859 p^{3} T^{4} + 402 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 838 T + 1394903 T^{2} - 671950004 T^{3} + 1394903 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 594 T + 357843 T^{2} - 156405492 T^{3} + 357843 p^{3} T^{4} + 594 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 2396 T + 3204249 T^{2} + 2882084008 T^{3} + 3204249 p^{3} T^{4} + 2396 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 170 T + 1042603 T^{2} - 206881916 T^{3} + 1042603 p^{3} T^{4} + 170 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 270 T + 2151919 T^{2} + 286220420 T^{3} + 2151919 p^{3} T^{4} + 270 p^{6} T^{5} + p^{9} T^{6} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.84478912715586378870444349204, −15.88180455081542440268242258296, −15.56194346425825622226188432957, −15.56017278704067734646266647191, −14.91646528533242250020355845294, −14.54293188176067705734753133801, −14.05477192401478735713806224842, −13.53903149732419800522825997454, −13.15157710861435480148132838771, −12.77482808723179505181256238678, −11.82587820016891111095247821777, −11.47528142904054404240864570025, −11.30398669605301579176000431340, −10.78651734741238135136874699464, −9.632473151912304259544737115312, −9.496516817172455924137662028802, −8.468797931306279695202548895799, −8.397324407657249551110908524756, −7.42880277196023059009466302805, −7.40099051835665882197576801088, −5.93382345550704463877322310005, −5.51674445038489297870992546733, −4.40891902220805417967654927496, −3.63824331318068623675418956800, −2.43281623707147080051052305557, 2.43281623707147080051052305557, 3.63824331318068623675418956800, 4.40891902220805417967654927496, 5.51674445038489297870992546733, 5.93382345550704463877322310005, 7.40099051835665882197576801088, 7.42880277196023059009466302805, 8.397324407657249551110908524756, 8.468797931306279695202548895799, 9.496516817172455924137662028802, 9.632473151912304259544737115312, 10.78651734741238135136874699464, 11.30398669605301579176000431340, 11.47528142904054404240864570025, 11.82587820016891111095247821777, 12.77482808723179505181256238678, 13.15157710861435480148132838771, 13.53903149732419800522825997454, 14.05477192401478735713806224842, 14.54293188176067705734753133801, 14.91646528533242250020355845294, 15.56017278704067734646266647191, 15.56194346425825622226188432957, 15.88180455081542440268242258296, 16.84478912715586378870444349204

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