Properties

Label 6-1856e3-1.1-c1e3-0-0
Degree $6$
Conductor $6393430016$
Sign $1$
Analytic cond. $3255.10$
Root an. cond. $3.84970$
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  + 2·3-s − 4·5-s + 2·11-s − 4·13-s − 8·15-s + 6·17-s − 4·19-s + 4·23-s + 4·25-s − 4·27-s − 3·29-s + 14·31-s + 4·33-s + 2·37-s − 8·39-s + 10·41-s − 6·43-s + 2·47-s − 21·49-s + 12·51-s − 8·55-s − 8·57-s + 8·59-s − 2·61-s + 16·65-s + 20·67-s + 8·69-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 0.603·11-s − 1.10·13-s − 2.06·15-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 4/5·25-s − 0.769·27-s − 0.557·29-s + 2.51·31-s + 0.696·33-s + 0.328·37-s − 1.28·39-s + 1.56·41-s − 0.914·43-s + 0.291·47-s − 3·49-s + 1.68·51-s − 1.07·55-s − 1.05·57-s + 1.04·59-s − 0.256·61-s + 1.98·65-s + 2.44·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{18} \cdot 29^{3}\)
Sign: $1$
Analytic conductor: \(3255.10\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1856} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{18} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.021445389\)
\(L(\frac12)\) \(\approx\) \(3.021445389\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
29$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 - 2 T + 4 T^{2} - 4 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + 4 T + 12 T^{2} + 6 p T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
7$C_2$ \( ( 1 + p T^{2} )^{3} \)
11$S_4\times C_2$ \( 1 - 2 T + 4 T^{2} + 36 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 20 T^{2} + 102 T^{3} + 20 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 4 T + 29 T^{2} + 120 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 14 T + 152 T^{2} - 936 T^{3} + 152 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 2 T + 79 T^{2} - 116 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 10 T + 59 T^{2} - 308 T^{3} + 59 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 92 T^{2} + 548 T^{3} + 92 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 2 T + 24 T^{2} + 264 T^{3} + 24 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 116 T^{2} - 58 T^{3} + 116 p T^{4} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 173 T^{2} - 928 T^{3} + 173 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 2 T + 83 T^{2} - 84 T^{3} + 83 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 20 T + 233 T^{2} - 2040 T^{3} + 233 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 12 T + 65 T^{2} + 8 T^{3} + 65 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 2 T + 123 T^{2} - 452 T^{3} + 123 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 30 T + 488 T^{2} - 5128 T^{3} + 488 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 32 T + 565 T^{2} - 6288 T^{3} + 565 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 10 T + 11 T^{2} + 1036 T^{3} + 11 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 14 T + 323 T^{2} + 2652 T^{3} + 323 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.275827121456768932018911749083, −7.981171681544970763320936435134, −7.72534603402878586194907581933, −7.54007372699774709849394215876, −7.41916950027733464999774978050, −6.80491798489944905710564185999, −6.77974615302055051431553914110, −6.37727271131036067042078777118, −6.04651957334448189316370596550, −6.00260828048511984479631256254, −5.16358704870627051564025232693, −5.02787933336758077516117835203, −5.02408656606041535181831260705, −4.47976621685674879321493778805, −4.11089923918800246471386863698, −4.02018853224252054978530500416, −3.52689088308962823922966992057, −3.28094142713394836570320414246, −3.19337989761556390878325289780, −2.72774958785343270474579145672, −2.25549755509276349133947459075, −2.12752472267340193791689713496, −1.45923667950737054077261786499, −0.72784945238847862396472887933, −0.54978540677904334919663387981, 0.54978540677904334919663387981, 0.72784945238847862396472887933, 1.45923667950737054077261786499, 2.12752472267340193791689713496, 2.25549755509276349133947459075, 2.72774958785343270474579145672, 3.19337989761556390878325289780, 3.28094142713394836570320414246, 3.52689088308962823922966992057, 4.02018853224252054978530500416, 4.11089923918800246471386863698, 4.47976621685674879321493778805, 5.02408656606041535181831260705, 5.02787933336758077516117835203, 5.16358704870627051564025232693, 6.00260828048511984479631256254, 6.04651957334448189316370596550, 6.37727271131036067042078777118, 6.77974615302055051431553914110, 6.80491798489944905710564185999, 7.41916950027733464999774978050, 7.54007372699774709849394215876, 7.72534603402878586194907581933, 7.981171681544970763320936435134, 8.275827121456768932018911749083

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