Properties

Label 6-4235e3-1.1-c1e3-0-2
Degree $6$
Conductor $75955677875$
Sign $-1$
Analytic cond. $38671.5$
Root an. cond. $5.81520$
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  + 3·2-s − 2·3-s + 4·4-s − 3·5-s − 6·6-s + 3·7-s + 4·8-s − 3·9-s − 9·10-s − 8·12-s − 2·13-s + 9·14-s + 6·15-s + 3·16-s − 9·18-s − 6·19-s − 12·20-s − 6·21-s − 10·23-s − 8·24-s + 6·25-s − 6·26-s + 10·27-s + 12·28-s + 10·29-s + 18·30-s − 10·31-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s − 2.44·6-s + 1.13·7-s + 1.41·8-s − 9-s − 2.84·10-s − 2.30·12-s − 0.554·13-s + 2.40·14-s + 1.54·15-s + 3/4·16-s − 2.12·18-s − 1.37·19-s − 2.68·20-s − 1.30·21-s − 2.08·23-s − 1.63·24-s + 6/5·25-s − 1.17·26-s + 1.92·27-s + 2.26·28-s + 1.85·29-s + 3.28·30-s − 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(5^{3} \cdot 7^{3} \cdot 11^{6}\)
Sign: $-1$
Analytic conductor: \(38671.5\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4235} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 5^{3} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad5$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good2$S_4\times C_2$ \( 1 - 3 T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
3$S_4\times C_2$ \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 17 T^{2} + 54 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 21 T^{2} - 2 T^{3} + 21 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 6 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 97 T^{2} + 480 T^{3} + 97 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 10 T + 113 T^{2} + 594 T^{3} + 113 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 16 T + 163 T^{2} + 1084 T^{3} + 163 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 87 T^{2} + 54 T^{3} + 87 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 2 T + 37 T^{2} - 440 T^{3} + 37 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 20 T + 251 T^{2} + 2038 T^{3} + 251 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 12 T + 179 T^{2} + 1276 T^{3} + 179 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 14 T + 3 p T^{2} - 1578 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 10 T + 123 T^{2} + 1282 T^{3} + 123 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 2 T + 89 T^{2} + 96 T^{3} + 89 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 24 T + 293 T^{2} + 2608 T^{3} + 293 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 4 T + 61 T^{2} + 394 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 8 T + 125 T^{2} + 1020 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 10 T + 133 T^{2} - 564 T^{3} + 133 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 20 T + 315 T^{2} - 3240 T^{3} + 315 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 99 T^{2} - 160 T^{3} + 99 p T^{4} + p^{3} T^{6} \)
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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

−7.58865313164770722728634366841, −7.41521569550597947372831559612, −7.37562977029193793833672257068, −7.00116642646110917256463750055, −6.57099044643292256822477397115, −6.35011428390453590773251296220, −6.33719938254107113860152692902, −5.95727993395610976695065466334, −5.64088101164823871091446039586, −5.57469812107072089090735828244, −5.00016759672718433075283058947, −4.96865993925329474311816956920, −4.88602809142225917595625630127, −4.65297363146386151934892208912, −4.32280712671900465423079312014, −4.11146135719352011891653453831, −3.74058465021985711254123288528, −3.45966285607318574460234225772, −3.45435998833542667408520947545, −2.87578162814776096265862687262, −2.66160670152634085717806283879, −2.20105613770269576807152234021, −1.74103237889578837106585752051, −1.73807617984567816411014982642, −1.05285008971948718870510312235, 0, 0, 0, 1.05285008971948718870510312235, 1.73807617984567816411014982642, 1.74103237889578837106585752051, 2.20105613770269576807152234021, 2.66160670152634085717806283879, 2.87578162814776096265862687262, 3.45435998833542667408520947545, 3.45966285607318574460234225772, 3.74058465021985711254123288528, 4.11146135719352011891653453831, 4.32280712671900465423079312014, 4.65297363146386151934892208912, 4.88602809142225917595625630127, 4.96865993925329474311816956920, 5.00016759672718433075283058947, 5.57469812107072089090735828244, 5.64088101164823871091446039586, 5.95727993395610976695065466334, 6.33719938254107113860152692902, 6.35011428390453590773251296220, 6.57099044643292256822477397115, 7.00116642646110917256463750055, 7.37562977029193793833672257068, 7.41521569550597947372831559612, 7.58865313164770722728634366841

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