Properties

Label 6-7800e3-1.1-c1e3-0-0
Degree $6$
Conductor $474552000000$
Sign $1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 7-s + 6·9-s + 5·11-s + 3·13-s − 9·17-s − 2·19-s + 3·21-s + 4·23-s − 10·27-s − 5·29-s + 5·31-s − 15·33-s + 6·37-s − 9·39-s − 13·47-s + 3·49-s + 27·51-s + 53-s + 6·57-s + 13·59-s + 11·61-s − 6·63-s − 25·67-s − 12·69-s + 2·71-s − 2·73-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s + 1.50·11-s + 0.832·13-s − 2.18·17-s − 0.458·19-s + 0.654·21-s + 0.834·23-s − 1.92·27-s − 0.928·29-s + 0.898·31-s − 2.61·33-s + 0.986·37-s − 1.44·39-s − 1.89·47-s + 3/7·49-s + 3.78·51-s + 0.137·53-s + 0.794·57-s + 1.69·59-s + 1.40·61-s − 0.755·63-s − 3.05·67-s − 1.44·69-s + 0.237·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(241610.\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.059572827\)
\(L(\frac12)\) \(\approx\) \(1.059572827\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + T - 2 T^{2} - 19 T^{3} - 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 5 T + 28 T^{2} - 107 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 2 T + 21 T^{2} + 136 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T + 37 T^{2} - 52 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 5 T + 42 T^{2} + 89 T^{3} + 42 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 88 T^{2} - 307 T^{3} + 88 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 87 T^{2} - 424 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 87 T^{2} + 44 T^{3} + 87 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 93 T^{2} - 44 T^{3} + 93 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 13 T + 122 T^{2} + 861 T^{3} + 122 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - T + 122 T^{2} - 9 T^{3} + 122 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 13 T + 158 T^{2} - 1173 T^{3} + 158 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 11 T + 170 T^{2} - 1079 T^{3} + 170 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 25 T + 396 T^{2} + 3803 T^{3} + 396 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 2 T + 73 T^{2} + 352 T^{3} + 73 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 2 T + 79 T^{2} - 344 T^{3} + 79 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 249 T^{2} - 1860 T^{3} + 249 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 15 T + 246 T^{2} - 2027 T^{3} + 246 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 16 T + 299 T^{2} - 2832 T^{3} + 299 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 14 T + 263 T^{2} + 2180 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.84030214290695697592810738560, −6.54523130366209625806727886228, −6.50632224340359983542609703050, −6.28151000040561662183816427258, −6.08960533366683173105032132290, −5.98514808863659217335385284837, −5.56015353867194372565392774434, −5.14705108993423152160153539752, −5.08261257586526528746151853632, −4.97232584937965596627538021390, −4.49441171111645072918124767353, −4.35949279828071887379529756808, −4.13559238269110737022208862734, −3.71395206035176505803930664235, −3.70423931910554467302953899005, −3.58865592329480564298293670290, −2.78448621832531156487853621400, −2.65589169370230981187220115398, −2.50657134113328661151702658932, −1.84295891356769524160335010379, −1.62145352389692138147839197307, −1.53369341376602720504193493637, −0.857419905429107983260192090025, −0.74892412139391629667838280183, −0.23763168854140627782972561740, 0.23763168854140627782972561740, 0.74892412139391629667838280183, 0.857419905429107983260192090025, 1.53369341376602720504193493637, 1.62145352389692138147839197307, 1.84295891356769524160335010379, 2.50657134113328661151702658932, 2.65589169370230981187220115398, 2.78448621832531156487853621400, 3.58865592329480564298293670290, 3.70423931910554467302953899005, 3.71395206035176505803930664235, 4.13559238269110737022208862734, 4.35949279828071887379529756808, 4.49441171111645072918124767353, 4.97232584937965596627538021390, 5.08261257586526528746151853632, 5.14705108993423152160153539752, 5.56015353867194372565392774434, 5.98514808863659217335385284837, 6.08960533366683173105032132290, 6.28151000040561662183816427258, 6.50632224340359983542609703050, 6.54523130366209625806727886228, 6.84030214290695697592810738560

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