Properties

Label 6-6160e3-1.1-c1e3-0-8
Degree $6$
Conductor $233744896000$
Sign $-1$
Analytic cond. $119007.$
Root an. cond. $7.01340$
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  − 2·3-s − 3·5-s + 3·7-s − 9-s + 3·11-s − 2·13-s + 6·15-s − 4·17-s − 6·19-s − 6·21-s − 2·23-s + 6·25-s + 6·27-s − 6·29-s + 6·31-s − 6·33-s − 9·35-s − 4·37-s + 4·39-s − 12·41-s + 10·43-s + 3·45-s − 12·47-s + 6·49-s + 8·51-s + 8·53-s − 9·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 1.13·7-s − 1/3·9-s + 0.904·11-s − 0.554·13-s + 1.54·15-s − 0.970·17-s − 1.37·19-s − 1.30·21-s − 0.417·23-s + 6/5·25-s + 1.15·27-s − 1.11·29-s + 1.07·31-s − 1.04·33-s − 1.52·35-s − 0.657·37-s + 0.640·39-s − 1.87·41-s + 1.52·43-s + 0.447·45-s − 1.75·47-s + 6/7·49-s + 1.12·51-s + 1.09·53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(119007.\)
Root analytic conductor: \(7.01340\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
11$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 35 T^{2} + 46 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 31 T^{2} + 70 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 6 T + 37 T^{2} + 140 T^{3} + 37 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 2 T + 53 T^{2} + 96 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
31$S_4\times C_2$ \( 1 - 6 T + 67 T^{2} - 218 T^{3} + 67 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 99 T^{2} + 260 T^{3} + 99 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 12 T + 133 T^{2} + 982 T^{3} + 133 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 77 T^{2} - 512 T^{3} + 77 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 12 T + 145 T^{2} + 1010 T^{3} + 145 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 8 T + 139 T^{2} - 700 T^{3} + 139 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 2 T - 5 T^{2} - 710 T^{3} - 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 22 T + 249 T^{2} + 2058 T^{3} + 249 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 6 T + 141 T^{2} - 560 T^{3} + 141 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 12 T + 133 T^{2} - 1512 T^{3} + 133 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 20 T + 327 T^{2} + 3082 T^{3} + 327 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 32 T + 537 T^{2} - 5772 T^{3} + 537 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 14 T + 285 T^{2} + 2236 T^{3} + 285 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 4 T + 251 T^{2} - 664 T^{3} + 251 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 4 T + 99 T^{2} - 1560 T^{3} + 99 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.49379789510451122581915965355, −7.16803828712592438400273175535, −6.90439910574245381054672512754, −6.88954559492186336378664122714, −6.35286852809047599018243157109, −6.31268869311300036962581402023, −6.29481968872842863479634363187, −5.61764413448872724858171705974, −5.52963934089536074915408746457, −5.50859115809739797985054492547, −4.90651111433906934923061444325, −4.77803431737844659170523619284, −4.54192711688217087117681061995, −4.44434050789582462950446322638, −4.14715839200490986527462785506, −3.93831998491024625371087071585, −3.33282062092290517319335201682, −3.27829343477334136814901091239, −3.25967460176208348156636914442, −2.32013302890219695624796532015, −2.31137766100320215347116757583, −2.15206026523181725862382305778, −1.60242112175783595127630363366, −1.12524496262892800779036826500, −1.00993013567478273050309231425, 0, 0, 0, 1.00993013567478273050309231425, 1.12524496262892800779036826500, 1.60242112175783595127630363366, 2.15206026523181725862382305778, 2.31137766100320215347116757583, 2.32013302890219695624796532015, 3.25967460176208348156636914442, 3.27829343477334136814901091239, 3.33282062092290517319335201682, 3.93831998491024625371087071585, 4.14715839200490986527462785506, 4.44434050789582462950446322638, 4.54192711688217087117681061995, 4.77803431737844659170523619284, 4.90651111433906934923061444325, 5.50859115809739797985054492547, 5.52963934089536074915408746457, 5.61764413448872724858171705974, 6.29481968872842863479634363187, 6.31268869311300036962581402023, 6.35286852809047599018243157109, 6.88954559492186336378664122714, 6.90439910574245381054672512754, 7.16803828712592438400273175535, 7.49379789510451122581915965355

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