Properties

Label 6-5733e3-1.1-c1e3-0-4
Degree $6$
Conductor $188428167837$
Sign $-1$
Analytic cond. $95935.0$
Root an. cond. $6.76596$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 8-s − 2·10-s − 2·11-s − 3·13-s − 16-s + 4·17-s + 4·19-s − 2·20-s + 2·22-s − 10·23-s − 8·25-s + 3·26-s − 24·29-s + 4·31-s + 32-s − 4·34-s − 4·38-s + 2·40-s + 2·41-s + 10·43-s + 2·44-s + 10·46-s − 8·47-s + 8·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.603·11-s − 0.832·13-s − 1/4·16-s + 0.970·17-s + 0.917·19-s − 0.447·20-s + 0.426·22-s − 2.08·23-s − 8/5·25-s + 0.588·26-s − 4.45·29-s + 0.718·31-s + 0.176·32-s − 0.685·34-s − 0.648·38-s + 0.316·40-s + 0.312·41-s + 1.52·43-s + 0.301·44-s + 1.47·46-s − 1.16·47-s + 1.13·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(95935.0\)
Root analytic conductor: \(6.76596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5733} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{6} \cdot 7^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
13$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
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

−7.61929158389835206292358108821, −7.45301200944173434401777577436, −7.41490125714622245671848529832, −6.74806846502437154940726323465, −6.73557186223943787482678231608, −6.04506516239010667841791313432, −6.00173115838023391012777872041, −5.90859225246227753217270002182, −5.72927753631409768565412057026, −5.49495758383364889825744542396, −5.06898064119564441732956838655, −5.03536941510938430234784295977, −4.61848616347178031920120262817, −4.25383830594107742839147924885, −4.11710879585163276589862533402, −3.81766842303496704797449335203, −3.37365015269456516225678205651, −3.35243017741884140470896179548, −3.06182567836368138698080706012, −2.35769717024745362005061856055, −2.18648840239592804521063851983, −2.00141128250161081989778808507, −1.91024563837143301616346394150, −1.21638248487755075255628957140, −1.05568239992044774063077333034, 0, 0, 0, 1.05568239992044774063077333034, 1.21638248487755075255628957140, 1.91024563837143301616346394150, 2.00141128250161081989778808507, 2.18648840239592804521063851983, 2.35769717024745362005061856055, 3.06182567836368138698080706012, 3.35243017741884140470896179548, 3.37365015269456516225678205651, 3.81766842303496704797449335203, 4.11710879585163276589862533402, 4.25383830594107742839147924885, 4.61848616347178031920120262817, 5.03536941510938430234784295977, 5.06898064119564441732956838655, 5.49495758383364889825744542396, 5.72927753631409768565412057026, 5.90859225246227753217270002182, 6.00173115838023391012777872041, 6.04506516239010667841791313432, 6.73557186223943787482678231608, 6.74806846502437154940726323465, 7.41490125714622245671848529832, 7.45301200944173434401777577436, 7.61929158389835206292358108821

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