Properties

Label 6-637e3-1.1-c1e3-0-2
Degree $6$
Conductor $258474853$
Sign $-1$
Analytic cond. $131.598$
Root an. cond. $2.25532$
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  − 2·2-s − 4·3-s + 2·4-s − 5·5-s + 8·6-s − 2·8-s + 9·9-s + 10·10-s − 4·11-s − 8·12-s + 3·13-s + 20·15-s − 4·17-s − 18·18-s − 7·19-s − 10·20-s + 8·22-s + 23-s + 8·24-s + 7·25-s − 6·26-s − 18·27-s − 7·29-s − 40·30-s + 3·31-s + 16·33-s + 8·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 4-s − 2.23·5-s + 3.26·6-s − 0.707·8-s + 3·9-s + 3.16·10-s − 1.20·11-s − 2.30·12-s + 0.832·13-s + 5.16·15-s − 0.970·17-s − 4.24·18-s − 1.60·19-s − 2.23·20-s + 1.70·22-s + 0.208·23-s + 1.63·24-s + 7/5·25-s − 1.17·26-s − 3.46·27-s − 1.29·29-s − 7.30·30-s + 0.538·31-s + 2.78·33-s + 1.37·34-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(7^{6} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(131.598\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 7^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad7 \( 1 \)
13$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 + p T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
3$S_4\times C_2$ \( 1 + 4 T + 7 T^{2} + 10 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + p T + 18 T^{2} + 43 T^{3} + 18 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 4 T + 3 p T^{2} + 86 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 49 T^{2} + 122 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 7 T + 54 T^{2} + 203 T^{3} + 54 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 28 T^{2} - 89 T^{3} + 28 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 7 T + 74 T^{2} + 403 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 52 T^{2} - 137 T^{3} + 52 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 10 T + 119 T^{2} + 658 T^{3} + 119 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 6 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 17 T + 230 T^{2} + 1745 T^{3} + 230 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 13 T + 198 T^{2} - 1369 T^{3} + 198 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 22 T + 321 T^{2} + 2848 T^{3} + 321 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 24 T + 343 T^{2} - 3152 T^{3} + 343 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 14 T + 165 T^{2} + 1228 T^{3} + 165 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 169 T^{2} - 374 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 5 T - 831 T^{3} + 5 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - T + 168 T^{2} - 257 T^{3} + 168 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 23 T + 376 T^{2} + 4021 T^{3} + 376 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 11 T + 212 T^{2} - 1979 T^{3} + 212 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 3 T + 220 T^{2} - 575 T^{3} + 220 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.886448827782354447207839573212, −9.738586663512547851786590594763, −9.138262503150044837639437170910, −9.092376386554425210593750565535, −8.499226119756723269213361127937, −8.372852030371903580352969069313, −8.192831151997316336892533342234, −7.900006604580244668542591018722, −7.42914455428864615921675834751, −7.11216675889706320106399394130, −7.01531503002281907134948596800, −6.76550769822222441770282968481, −6.22389698425433008111806872943, −6.01080538658186059675657642303, −5.53414056094829906550762555246, −5.50561498754813863688883229204, −4.76321301533434046171986553243, −4.66124219223746332786962196405, −4.17936796705809210750168451694, −4.00510973766594395050002517318, −3.47344476654452636802043916797, −3.14489727817398470301905971745, −2.17913938645567897664155777485, −1.94168055883410583124853330460, −1.10596898171468342754065453783, 0, 0, 0, 1.10596898171468342754065453783, 1.94168055883410583124853330460, 2.17913938645567897664155777485, 3.14489727817398470301905971745, 3.47344476654452636802043916797, 4.00510973766594395050002517318, 4.17936796705809210750168451694, 4.66124219223746332786962196405, 4.76321301533434046171986553243, 5.50561498754813863688883229204, 5.53414056094829906550762555246, 6.01080538658186059675657642303, 6.22389698425433008111806872943, 6.76550769822222441770282968481, 7.01531503002281907134948596800, 7.11216675889706320106399394130, 7.42914455428864615921675834751, 7.900006604580244668542591018722, 8.192831151997316336892533342234, 8.372852030371903580352969069313, 8.499226119756723269213361127937, 9.092376386554425210593750565535, 9.138262503150044837639437170910, 9.738586663512547851786590594763, 9.886448827782354447207839573212

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