Properties

Label 6-637e3-1.1-c1e3-0-1
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 $0$

Origins

Origins of factors

Downloads

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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: \(0\)
Selberg data: \((6,\ 7^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.843265871\)
\(L(\frac12)\) \(\approx\) \(3.843265871\)
\(L(\frac{3}{2})\) 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

−9.529867127642993661590570600186, −9.182642743920545154488125772139, −8.889183321749360410690022288154, −8.785443725698670533903548305581, −8.431020605800283352557996123618, −8.000652481642021648717183770220, −7.67575127913523731317324166520, −7.44075378745038186326811013386, −7.29079128890212255960207671977, −7.11318411145537296109878140346, −6.62127784116218580160991895086, −5.89252159464686171875496880292, −5.80032554366248199096050883347, −5.43142123789935159051410651735, −5.38399324176441662561470680126, −4.73813545716517336757014977831, −4.24210765989283840943816093661, −3.73708896342617125646146836606, −3.18549583824662157349818568266, −3.13580267910941229711257118119, −2.35238640756927560759022767398, −2.28941462341848929539677406425, −2.14382562422290513776396535094, −1.47501913933754980833286348448, −0.821424707671999522381674432919, 0.821424707671999522381674432919, 1.47501913933754980833286348448, 2.14382562422290513776396535094, 2.28941462341848929539677406425, 2.35238640756927560759022767398, 3.13580267910941229711257118119, 3.18549583824662157349818568266, 3.73708896342617125646146836606, 4.24210765989283840943816093661, 4.73813545716517336757014977831, 5.38399324176441662561470680126, 5.43142123789935159051410651735, 5.80032554366248199096050883347, 5.89252159464686171875496880292, 6.62127784116218580160991895086, 7.11318411145537296109878140346, 7.29079128890212255960207671977, 7.44075378745038186326811013386, 7.67575127913523731317324166520, 8.000652481642021648717183770220, 8.431020605800283352557996123618, 8.785443725698670533903548305581, 8.889183321749360410690022288154, 9.182642743920545154488125772139, 9.529867127642993661590570600186

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