Properties

Label 6-2166e3-1.1-c3e3-0-5
Degree $6$
Conductor $10161910296$
Sign $-1$
Analytic cond. $2.08724\times 10^{6}$
Root an. cond. $11.3047$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 9·3-s + 24·4-s + 2·5-s + 54·6-s − 17·7-s + 80·8-s + 54·9-s + 12·10-s − 52·11-s + 216·12-s − 75·13-s − 102·14-s + 18·15-s + 240·16-s − 48·17-s + 324·18-s + 48·20-s − 153·21-s − 312·22-s − 238·23-s + 720·24-s − 71·25-s − 450·26-s + 270·27-s − 408·28-s + 8·29-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3·4-s + 0.178·5-s + 3.67·6-s − 0.917·7-s + 3.53·8-s + 2·9-s + 0.379·10-s − 1.42·11-s + 5.19·12-s − 1.60·13-s − 1.94·14-s + 0.309·15-s + 15/4·16-s − 0.684·17-s + 4.24·18-s + 0.536·20-s − 1.58·21-s − 3.02·22-s − 2.15·23-s + 6.12·24-s − 0.567·25-s − 3.39·26-s + 1.92·27-s − 2.75·28-s + 0.0512·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 19^{6}\)
Sign: $-1$
Analytic conductor: \(2.08724\times 10^{6}\)
Root analytic conductor: \(11.3047\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 19^{6} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{3} \)
3$C_1$ \( ( 1 - p T )^{3} \)
19 \( 1 \)
good5$S_4\times C_2$ \( 1 - 2 T + 3 p^{2} T^{2} - 1868 T^{3} + 3 p^{5} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 17 T + 88 T^{2} - 6791 T^{3} + 88 p^{3} T^{4} + 17 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 52 T + 3105 T^{2} + 105736 T^{3} + 3105 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 75 T + 570 p T^{2} + 316343 T^{3} + 570 p^{4} T^{4} + 75 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 48 T + 13011 T^{2} + 482016 T^{3} + 13011 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 238 T + 13449 T^{2} - 313172 T^{3} + 13449 p^{3} T^{4} + 238 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 8 T + 30351 T^{2} - 83792 T^{3} + 30351 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 107 T + 75368 T^{2} + 4940027 T^{3} + 75368 p^{3} T^{4} + 107 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 305 T + 26786 T^{2} + 1997981 T^{3} + 26786 p^{3} T^{4} + 305 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 16 T + 40203 T^{2} + 9213088 T^{3} + 40203 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 331 T + 154580 T^{2} + 55755235 T^{3} + 154580 p^{3} T^{4} + 331 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 766 T + 402777 T^{2} + 138860332 T^{3} + 402777 p^{3} T^{4} + 766 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 118 T + 228651 T^{2} + 5659004 T^{3} + 228651 p^{3} T^{4} - 118 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 936 T + 766257 T^{2} + 352800000 T^{3} + 766257 p^{3} T^{4} + 936 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 399 T + 471354 T^{2} + 102958675 T^{3} + 471354 p^{3} T^{4} + 399 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 61 T + 714028 T^{2} + 959687 p T^{3} + 714028 p^{3} T^{4} + 61 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 974 T + 1095153 T^{2} + 680220716 T^{3} + 1095153 p^{3} T^{4} + 974 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 91 T + 598750 T^{2} + 96409345 T^{3} + 598750 p^{3} T^{4} - 91 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 321 T + 1051536 T^{2} - 380263561 T^{3} + 1051536 p^{3} T^{4} - 321 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 2148 T + 2987025 T^{2} + 2667812568 T^{3} + 2987025 p^{3} T^{4} + 2148 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 1116 T + 2360067 T^{2} + 1584528840 T^{3} + 2360067 p^{3} T^{4} + 1116 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 1382 T + 3339119 T^{2} + 2601906260 T^{3} + 3339119 p^{3} T^{4} + 1382 p^{6} T^{5} + p^{9} T^{6} \)
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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

−7.989650497026993419497424191176, −7.72733973931323586713867158302, −7.44068747962895723673206560638, −7.34919324829950716935968061536, −7.00858391644437634667392949510, −6.81903378770549736575854337293, −6.36921030693616963690089516876, −6.12443891193494000997479251201, −6.11775613268192312611817653965, −5.57506275932175800722478683524, −5.17560852112574304966106958961, −5.12109511745213571704598672463, −4.88803506370909219650598583290, −4.35797151214545110113316873409, −4.21449279135148404476866594604, −4.05811785024509853956879556387, −3.55080273956737789592883180762, −3.33748458512727510122371709184, −2.94575367698954770313552622112, −2.88802149649476282285472712290, −2.59324039512907571203166969239, −2.18665981137565323238422930099, −1.80203237118031250041125231170, −1.78200284540918397062871622277, −1.32274174276966197038975451196, 0, 0, 0, 1.32274174276966197038975451196, 1.78200284540918397062871622277, 1.80203237118031250041125231170, 2.18665981137565323238422930099, 2.59324039512907571203166969239, 2.88802149649476282285472712290, 2.94575367698954770313552622112, 3.33748458512727510122371709184, 3.55080273956737789592883180762, 4.05811785024509853956879556387, 4.21449279135148404476866594604, 4.35797151214545110113316873409, 4.88803506370909219650598583290, 5.12109511745213571704598672463, 5.17560852112574304966106958961, 5.57506275932175800722478683524, 6.11775613268192312611817653965, 6.12443891193494000997479251201, 6.36921030693616963690089516876, 6.81903378770549736575854337293, 7.00858391644437634667392949510, 7.34919324829950716935968061536, 7.44068747962895723673206560638, 7.72733973931323586713867158302, 7.989650497026993419497424191176

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