Properties

Label 6-5586e3-1.1-c1e3-0-0
Degree $6$
Conductor $174302170056$
Sign $1$
Analytic cond. $88743.0$
Root an. cond. $6.67865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·3-s + 6·4-s − 5-s − 9·6-s − 10·8-s + 6·9-s + 3·10-s + 7·11-s + 18·12-s − 2·13-s − 3·15-s + 15·16-s + 10·17-s − 18·18-s + 3·19-s − 6·20-s − 21·22-s + 10·23-s − 30·24-s − 7·25-s + 6·26-s + 10·27-s − 7·29-s + 9·30-s + 11·31-s − 21·32-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 3·4-s − 0.447·5-s − 3.67·6-s − 3.53·8-s + 2·9-s + 0.948·10-s + 2.11·11-s + 5.19·12-s − 0.554·13-s − 0.774·15-s + 15/4·16-s + 2.42·17-s − 4.24·18-s + 0.688·19-s − 1.34·20-s − 4.47·22-s + 2.08·23-s − 6.12·24-s − 7/5·25-s + 1.17·26-s + 1.92·27-s − 1.29·29-s + 1.64·30-s + 1.97·31-s − 3.71·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(88743.0\)
Root analytic conductor: \(6.67865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.359338567\)
\(L(\frac12)\) \(\approx\) \(5.359338567\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
3$C_1$ \( ( 1 - T )^{3} \)
7 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 + T + 8 T^{2} + T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 7 T + 32 T^{2} - 111 T^{3} + 32 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 19 T^{2} + 60 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 10 T + 77 T^{2} - 346 T^{3} + 77 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 10 T + 77 T^{2} - 378 T^{3} + 77 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 7 T + 62 T^{2} + 379 T^{3} + 62 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 11 T + 56 T^{2} - 203 T^{3} + 56 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + p T^{2} - 66 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 119 T^{2} - 158 T^{3} + 119 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 103 T^{2} + 534 T^{3} + 103 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 109 T^{2} - 32 T^{3} + 109 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - T + 138 T^{2} - 77 T^{3} + 138 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 3 T + 64 T^{2} - 119 T^{3} + 64 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 22 T + 315 T^{2} - 2916 T^{3} + 315 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 20 T + 305 T^{2} + 2728 T^{3} + 305 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 10 T + 239 T^{2} + 1426 T^{3} + 239 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 135 T^{2} + 1064 T^{3} + 135 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 3 T + 40 T^{2} + 775 T^{3} + 40 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 19 T + 340 T^{2} + 3275 T^{3} + 340 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 199 T^{2} - 26 p T^{3} + 199 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 13 T + 214 T^{2} + 1683 T^{3} + 214 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.31957564999929682331865177676, −7.27165115950431155055786667577, −7.23829081288897085807624629691, −6.62123739212025411460060750189, −6.30006713764108327849879910423, −6.19242403315112269366831208013, −6.05052575558557700185388465020, −5.45759549753313833298626440137, −5.45280530218147021322851923804, −5.02425033079984745963171859811, −4.60684330678310338234788265749, −4.35840425583088159203668436464, −4.08523622814964570917156629752, −3.69471523746809040834783091350, −3.54406372946654883826976582487, −3.19132344685762194322243340073, −3.05492976554083074258046037736, −2.83281474586753987192716584875, −2.56387113734584011958488160632, −1.82323329339745165800079125005, −1.75100243115147256867253388534, −1.65292107053386720081871543464, −1.05873106433594582701980477755, −0.851046464014087422326047887310, −0.57112917761301902685645861892, 0.57112917761301902685645861892, 0.851046464014087422326047887310, 1.05873106433594582701980477755, 1.65292107053386720081871543464, 1.75100243115147256867253388534, 1.82323329339745165800079125005, 2.56387113734584011958488160632, 2.83281474586753987192716584875, 3.05492976554083074258046037736, 3.19132344685762194322243340073, 3.54406372946654883826976582487, 3.69471523746809040834783091350, 4.08523622814964570917156629752, 4.35840425583088159203668436464, 4.60684330678310338234788265749, 5.02425033079984745963171859811, 5.45280530218147021322851923804, 5.45759549753313833298626440137, 6.05052575558557700185388465020, 6.19242403315112269366831208013, 6.30006713764108327849879910423, 6.62123739212025411460060750189, 7.23829081288897085807624629691, 7.27165115950431155055786667577, 7.31957564999929682331865177676

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