Properties

Label 6-231e3-1.1-c1e3-0-1
Degree $6$
Conductor $12326391$
Sign $1$
Analytic cond. $6.27577$
Root an. cond. $1.35814$
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  + 2·2-s + 3·3-s + 2·4-s + 4·5-s + 6·6-s − 3·7-s + 8-s + 6·9-s + 8·10-s − 3·11-s + 6·12-s − 4·13-s − 6·14-s + 12·15-s − 4·16-s + 8·17-s + 12·18-s − 8·19-s + 8·20-s − 9·21-s − 6·22-s + 10·23-s + 3·24-s + 8·25-s − 8·26-s + 10·27-s − 6·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 4-s + 1.78·5-s + 2.44·6-s − 1.13·7-s + 0.353·8-s + 2·9-s + 2.52·10-s − 0.904·11-s + 1.73·12-s − 1.10·13-s − 1.60·14-s + 3.09·15-s − 16-s + 1.94·17-s + 2.82·18-s − 1.83·19-s + 1.78·20-s − 1.96·21-s − 1.27·22-s + 2.08·23-s + 0.612·24-s + 8/5·25-s − 1.56·26-s + 1.92·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12326391 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12326391 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(12326391\)    =    \(3^{3} \cdot 7^{3} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(6.27577\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{231} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 12326391,\ (\ :1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.536096336\)
\(L(\frac12)\) \(\approx\) \(6.536096336\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
11$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 - p T + p T^{2} - T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 14 T^{3} + 8 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 12 T^{2} + 10 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 8 T + 11 T^{2} + 56 T^{3} + 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 8 T + 72 T^{2} + 308 T^{3} + 72 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 10 T + 81 T^{2} - 396 T^{3} + 81 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 60 T^{2} + 138 T^{3} + 60 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 17 T^{2} - 132 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 68 T^{2} + 106 T^{3} + 68 p T^{4} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 14 T + 163 T^{2} - 1116 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 14 T + 85 T^{2} + 356 T^{3} + 85 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 80 T^{2} + 32 T^{3} + 80 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 143 T^{2} + 8 T^{3} + 143 p T^{4} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 120 T^{2} - 52 T^{3} + 120 p T^{4} + p^{3} T^{6} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 + 4 T + 116 T^{2} + 300 T^{3} + 116 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 12 T + 197 T^{2} + 1448 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 20 T + 320 T^{2} + 3054 T^{3} + 320 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 221 T^{2} - 1640 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 117 T^{2} - 500 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 26 T + 407 T^{2} - 4300 T^{3} + 407 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 4 T + 171 T^{2} + 544 T^{3} + 171 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−10.64145954225033776958527833627, −10.52997061934434268445770386664, −10.29749350306401942868080377740, −9.862077681512781941673610626796, −9.586722059945123837281668420209, −9.331081393855998552863132008413, −9.074704268752283185878023731561, −8.617384613230567924173898549112, −8.497173234860566319854544466412, −7.60328759004254524416029840779, −7.47539022279127247059199233696, −7.30779760444084841078467261484, −6.69435066610836823740758647067, −6.33252302571214420850142548681, −6.00493496158892074443379435471, −5.69653518869871741849245878104, −5.08278705547917041221747902990, −4.74820725735497277064419562229, −4.61930078937300391216756519466, −3.59901998740036571373751238840, −3.57320586066665000755186061480, −2.79494867967071850087724802645, −2.67958341900950104318063369812, −2.25819396769315172970158731875, −1.57297212451302457402103914109, 1.57297212451302457402103914109, 2.25819396769315172970158731875, 2.67958341900950104318063369812, 2.79494867967071850087724802645, 3.57320586066665000755186061480, 3.59901998740036571373751238840, 4.61930078937300391216756519466, 4.74820725735497277064419562229, 5.08278705547917041221747902990, 5.69653518869871741849245878104, 6.00493496158892074443379435471, 6.33252302571214420850142548681, 6.69435066610836823740758647067, 7.30779760444084841078467261484, 7.47539022279127247059199233696, 7.60328759004254524416029840779, 8.497173234860566319854544466412, 8.617384613230567924173898549112, 9.074704268752283185878023731561, 9.331081393855998552863132008413, 9.586722059945123837281668420209, 9.862077681512781941673610626796, 10.29749350306401942868080377740, 10.52997061934434268445770386664, 10.64145954225033776958527833627

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