Properties

Label 6-6422e3-1.1-c1e3-0-11
Degree $6$
Conductor $264856663448$
Sign $-1$
Analytic cond. $134847.$
Root an. cond. $7.16100$
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 + 6·4-s + 2·5-s − 6·6-s − 7-s + 10·8-s − 2·9-s + 6·10-s − 8·11-s − 12·12-s − 3·14-s − 4·15-s + 15·16-s − 2·17-s − 6·18-s + 3·19-s + 12·20-s + 2·21-s − 24·22-s + 2·23-s − 20·24-s − 8·25-s + 9·27-s − 6·28-s − 14·29-s − 12·30-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.15·3-s + 3·4-s + 0.894·5-s − 2.44·6-s − 0.377·7-s + 3.53·8-s − 2/3·9-s + 1.89·10-s − 2.41·11-s − 3.46·12-s − 0.801·14-s − 1.03·15-s + 15/4·16-s − 0.485·17-s − 1.41·18-s + 0.688·19-s + 2.68·20-s + 0.436·21-s − 5.11·22-s + 0.417·23-s − 4.08·24-s − 8/5·25-s + 1.73·27-s − 1.13·28-s − 2.59·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 13^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 13^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 13^{6} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(134847.\)
Root analytic conductor: \(7.16100\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6422} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 13^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2$C_1$ \( ( 1 - T )^{3} \)
13 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good3$A_4\times C_2$ \( 1 + 2 T + 2 p T^{2} + 7 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 - 2 T + 12 T^{2} - 3 p T^{3} + 12 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + T + 17 T^{2} + 15 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 8 T + 50 T^{2} + 181 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 2 T + 40 T^{2} - 87 T^{3} + 40 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 14 T + 122 T^{2} + 709 T^{3} + 122 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 5 T + 45 T^{2} + 383 T^{3} + 45 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 59 T^{2} + 104 T^{3} + 59 p T^{4} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 15 T + 185 T^{2} - 1303 T^{3} + 185 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 5 T + 120 T^{2} + 425 T^{3} + 120 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 11 T + 125 T^{2} + 1039 T^{3} + 125 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 5 T + 111 T^{2} - 603 T^{3} + 111 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 4 T + 48 T^{2} + 471 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 2 T + 154 T^{2} - 161 T^{3} + 154 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 6 T + 161 T^{2} - 812 T^{3} + 161 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{3} \)
73$A_4\times C_2$ \( 1 - 15 T + 86 T^{2} - 443 T^{3} + 86 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 2 T + 130 T^{2} + 545 T^{3} + 130 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 5 T - 7 T^{2} + 995 T^{3} - 7 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 27 T + 471 T^{2} + 5119 T^{3} + 471 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 20 T + 407 T^{2} + 4080 T^{3} + 407 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.56318903976664310555946925092, −6.88852847107395876350750786855, −6.87817383565792900371852961096, −6.61272495823320514114820804484, −6.22898555292719715285877859988, −6.07484160047375323580754851039, −6.00271959262572494582884398875, −5.52099739220083971780658402990, −5.46865821155644263420166333982, −5.45384990867830813304577423821, −5.11100453771829539770035009467, −5.01360896241448717963976931614, −4.81341827676471401596396602021, −4.22024787551148334093741166644, −4.04887383461729504565321801256, −3.81491596962926872572071942873, −3.58759907322102110836898582188, −3.08759137113016327027651779366, −3.03591293896983949700267313058, −2.58176134129582938742176235720, −2.45795121123072601812206038386, −2.29700479233755829451569964032, −1.79547518476637143584703761611, −1.47724742401769061153813959609, −1.18820684039106953511934093974, 0, 0, 0, 1.18820684039106953511934093974, 1.47724742401769061153813959609, 1.79547518476637143584703761611, 2.29700479233755829451569964032, 2.45795121123072601812206038386, 2.58176134129582938742176235720, 3.03591293896983949700267313058, 3.08759137113016327027651779366, 3.58759907322102110836898582188, 3.81491596962926872572071942873, 4.04887383461729504565321801256, 4.22024787551148334093741166644, 4.81341827676471401596396602021, 5.01360896241448717963976931614, 5.11100453771829539770035009467, 5.45384990867830813304577423821, 5.46865821155644263420166333982, 5.52099739220083971780658402990, 6.00271959262572494582884398875, 6.07484160047375323580754851039, 6.22898555292719715285877859988, 6.61272495823320514114820804484, 6.87817383565792900371852961096, 6.88852847107395876350750786855, 7.56318903976664310555946925092

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