Properties

Label 6-825e3-1.1-c1e3-0-0
Degree $6$
Conductor $561515625$
Sign $1$
Analytic cond. $285.886$
Root an. cond. $2.56664$
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 − 3·3-s + 3·4-s + 6·6-s + 3·7-s − 4·8-s + 6·9-s + 3·11-s − 9·12-s + 5·13-s − 6·14-s + 3·16-s − 4·17-s − 12·18-s − 19-s − 9·21-s − 6·22-s + 12·24-s − 10·26-s − 10·27-s + 9·28-s + 2·29-s + 17·31-s − 6·32-s − 9·33-s + 8·34-s + 18·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 3/2·4-s + 2.44·6-s + 1.13·7-s − 1.41·8-s + 2·9-s + 0.904·11-s − 2.59·12-s + 1.38·13-s − 1.60·14-s + 3/4·16-s − 0.970·17-s − 2.82·18-s − 0.229·19-s − 1.96·21-s − 1.27·22-s + 2.44·24-s − 1.96·26-s − 1.92·27-s + 1.70·28-s + 0.371·29-s + 3.05·31-s − 1.06·32-s − 1.56·33-s + 1.37·34-s + 3·36-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(285.886\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9727601301\)
\(L(\frac12)\) \(\approx\) \(0.9727601301\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
good2$D_{6}$ \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 3 T + 2 p T^{2} - 25 T^{3} + 2 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 5 T + 3 p T^{2} - 122 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 26 T^{2} + 158 T^{3} + 26 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + T + 2 p T^{2} + 63 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 26 T^{2} - 58 T^{3} + 26 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 63 T^{2} - 132 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 17 T + 181 T^{2} - 1190 T^{3} + 181 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 36 T^{2} + 34 T^{3} + 36 p T^{4} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 4 T + 122 T^{2} - 326 T^{3} + 122 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 17 T + 97 T^{2} - 362 T^{3} + 97 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 10 T + 166 T^{2} + 948 T^{3} + 166 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 11 T^{2} - 188 T^{3} + 11 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 146 T^{2} + 572 T^{3} + 146 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 3 T + 95 T^{2} + 10 p T^{3} + 95 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 7 T + 9 T^{2} + 650 T^{3} + 9 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 26 T + 430 T^{2} - 4272 T^{3} + 430 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 175 T^{2} + 988 T^{3} + 175 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 6 T + 200 T^{2} - 736 T^{3} + 200 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 33 T^{2} + 4 p T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 167 T^{2} - 28 T^{3} + 167 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 29 T + 366 T^{2} - 3473 T^{3} + 366 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.317746928907326830723139025520, −8.775098137802566346648366262963, −8.482881437935347409914589599000, −8.301945235509049051971418287809, −8.036427216060085430202696456338, −7.70204594248208799903047377716, −7.46945845532189984358203724852, −6.84558157470127505151102660516, −6.82845815205885481660548654473, −6.38629343028157446917187768431, −6.28078342388598376042465698297, −6.09479694912721430877460294668, −5.73076258370924447681368461962, −5.18903554245840169965313175376, −4.80672537665227115967301278157, −4.68475121433303800236728927336, −4.27348728453076285006260020300, −3.95701434672746778984827673068, −3.45960082152895495384041287946, −2.92733427759953433024893904218, −2.31824531880050942501510202791, −1.91348772437630928259383586149, −1.48861736074664494015328279834, −0.923072317473441397802132256497, −0.65095342717078505375832497646, 0.65095342717078505375832497646, 0.923072317473441397802132256497, 1.48861736074664494015328279834, 1.91348772437630928259383586149, 2.31824531880050942501510202791, 2.92733427759953433024893904218, 3.45960082152895495384041287946, 3.95701434672746778984827673068, 4.27348728453076285006260020300, 4.68475121433303800236728927336, 4.80672537665227115967301278157, 5.18903554245840169965313175376, 5.73076258370924447681368461962, 6.09479694912721430877460294668, 6.28078342388598376042465698297, 6.38629343028157446917187768431, 6.82845815205885481660548654473, 6.84558157470127505151102660516, 7.46945845532189984358203724852, 7.70204594248208799903047377716, 8.036427216060085430202696456338, 8.301945235509049051971418287809, 8.482881437935347409914589599000, 8.775098137802566346648366262963, 9.317746928907326830723139025520

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