Properties

Label 6-91e3-1.1-c1e3-0-0
Degree $6$
Conductor $753571$
Sign $1$
Analytic cond. $0.383667$
Root an. cond. $0.852431$
Motivic weight $1$
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  + 2-s − 2·3-s − 4-s + 2·5-s − 2·6-s − 3·7-s − 8-s + 9-s + 2·10-s + 2·11-s + 2·12-s + 3·13-s − 3·14-s − 4·15-s − 16-s + 4·17-s + 18-s − 4·19-s − 2·20-s + 6·21-s + 2·22-s + 10·23-s + 2·24-s − 8·25-s + 3·26-s + 3·28-s + 24·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.816·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.603·11-s + 0.577·12-s + 0.832·13-s − 0.801·14-s − 1.03·15-s − 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 1.30·21-s + 0.426·22-s + 2.08·23-s + 0.408·24-s − 8/5·25-s + 0.588·26-s + 0.566·28-s + 4.45·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 753571 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 753571 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(753571\)    =    \(7^{3} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(0.383667\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{91} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 753571,\ (\ :1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8395767304\)
\(L(\frac12)\) \(\approx\) \(0.8395767304\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( ( 1 + T )^{3} \)
13$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
3$S_4\times C_2$ \( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - 2 T + 12 T^{2} - 18 T^{3} + 12 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 27 T^{2} - 36 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 4 T + 58 T^{2} + 148 T^{3} + 58 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 10 T + 70 T^{2} - 324 T^{3} + 70 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 4 T + 74 T^{2} + 264 T^{3} + 74 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 53 T^{2} - 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 95 T^{2} - 172 T^{3} + 95 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 8 T + 62 T^{2} + 208 T^{3} + 62 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 8 T + 124 T^{2} - 870 T^{3} + 124 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 4 T + 21 T^{2} - 216 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 + 12 T + 77 T^{2} + 632 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 6 T + 191 T^{2} + 868 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 120 T^{2} + 1186 T^{3} + 120 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 242 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T - 22 T^{2} - 1276 T^{3} - 22 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 172 T^{2} + 66 T^{3} + 172 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 10 T + 320 T^{2} + 1962 T^{3} + 320 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−12.80973583034402151499416733291, −12.48136097530947487315393465072, −11.94216493392605955822877387310, −11.59086863743711495577045499367, −11.58422510339649888894089558692, −10.64055173704041571584458917132, −10.60099224119543009400133905705, −10.18969239618582243369301667974, −9.895812578457702131950705188115, −9.345127097157821584327934150744, −8.948365161903600243579050520670, −8.764766573874037691399016881034, −8.219521117528238630060402005881, −7.55210189783927525293762275792, −6.96372729183181824285675067939, −6.58542708455169737072582918708, −6.22024670683491927018005029540, −5.76443864748167928645575928550, −5.71590372024041195022493422624, −4.88149030753689217866982759788, −4.44789041868343238406134685437, −4.12066848872037228013527344962, −3.15547340691232350774513411813, −2.79185347271348457784021644014, −1.24925895028719379559500876429, 1.24925895028719379559500876429, 2.79185347271348457784021644014, 3.15547340691232350774513411813, 4.12066848872037228013527344962, 4.44789041868343238406134685437, 4.88149030753689217866982759788, 5.71590372024041195022493422624, 5.76443864748167928645575928550, 6.22024670683491927018005029540, 6.58542708455169737072582918708, 6.96372729183181824285675067939, 7.55210189783927525293762275792, 8.219521117528238630060402005881, 8.764766573874037691399016881034, 8.948365161903600243579050520670, 9.345127097157821584327934150744, 9.895812578457702131950705188115, 10.18969239618582243369301667974, 10.60099224119543009400133905705, 10.64055173704041571584458917132, 11.58422510339649888894089558692, 11.59086863743711495577045499367, 11.94216493392605955822877387310, 12.48136097530947487315393465072, 12.80973583034402151499416733291

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