Properties

Label 6-2850e3-1.1-c1e3-0-2
Degree $6$
Conductor $23149125000$
Sign $1$
Analytic cond. $11785.9$
Root an. cond. $4.77046$
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 − 9·6-s + 10·8-s + 6·9-s + 8·11-s − 18·12-s + 4·13-s + 15·16-s + 2·17-s + 18·18-s + 3·19-s + 24·22-s − 8·23-s − 30·24-s + 12·26-s − 10·27-s + 10·29-s + 21·32-s − 24·33-s + 6·34-s + 36·36-s + 4·37-s + 9·38-s − 12·39-s + 4·41-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s + 3.53·8-s + 2·9-s + 2.41·11-s − 5.19·12-s + 1.10·13-s + 15/4·16-s + 0.485·17-s + 4.24·18-s + 0.688·19-s + 5.11·22-s − 1.66·23-s − 6.12·24-s + 2.35·26-s − 1.92·27-s + 1.85·29-s + 3.71·32-s − 4.17·33-s + 1.02·34-s + 6·36-s + 0.657·37-s + 1.45·38-s − 1.92·39-s + 0.624·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(11785.9\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(17.32501488\)
\(L(\frac12)\) \(\approx\) \(17.32501488\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 8 T + 41 T^{2} - 160 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 4 T + 23 T^{2} - 72 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 31 T^{2} - 60 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 8 T + 77 T^{2} + 352 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 10 T + 107 T^{2} - 572 T^{3} + 107 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 4 T + 43 T^{2} + 72 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 5 T^{2} - 244 T^{3} + 5 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 149 T^{2} - 736 T^{3} + 149 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 - 10 T + 173 T^{2} - 1172 T^{3} + 173 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 20 T + 249 T^{2} - 2360 T^{3} + 249 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 20 T + 261 T^{2} - 2520 T^{3} + 261 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 8 T + 155 T^{2} - 912 T^{3} + 155 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 221 T^{2} - 16 T^{3} + 221 p T^{4} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 4 T + 73 T^{2} + 504 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 24 T + 443 T^{2} - 4672 T^{3} + 443 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 22 T + 439 T^{2} - 4564 T^{3} + 439 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.66336827210750210261875024648, −7.28941650460627868143686261376, −7.04385114077478658360195358489, −6.70050469182400468847362832832, −6.47473306744498430484430295554, −6.47082808670588982150472077549, −6.17861540134138408290630075127, −5.95156157584620285987864921908, −5.86342731974160431804891755937, −5.44789255013430167252404435544, −4.99160121048984436468009542127, −4.90876453670553712726802206299, −4.79888333972843455806985589619, −4.26807159652247261969913956316, −4.17544800644602202315273018135, −3.75761581483371301093945822392, −3.56872708393186299936335304581, −3.35801230543915315290564186352, −3.23510892995052702924737950843, −2.21991208521503445339808477558, −2.07330895612808480526761291781, −2.04476563802979756023364690474, −1.05493677737871007217103639281, −1.03984464621972180411597826029, −0.819481846713935859873666670801, 0.819481846713935859873666670801, 1.03984464621972180411597826029, 1.05493677737871007217103639281, 2.04476563802979756023364690474, 2.07330895612808480526761291781, 2.21991208521503445339808477558, 3.23510892995052702924737950843, 3.35801230543915315290564186352, 3.56872708393186299936335304581, 3.75761581483371301093945822392, 4.17544800644602202315273018135, 4.26807159652247261969913956316, 4.79888333972843455806985589619, 4.90876453670553712726802206299, 4.99160121048984436468009542127, 5.44789255013430167252404435544, 5.86342731974160431804891755937, 5.95156157584620285987864921908, 6.17861540134138408290630075127, 6.47082808670588982150472077549, 6.47473306744498430484430295554, 6.70050469182400468847362832832, 7.04385114077478658360195358489, 7.28941650460627868143686261376, 7.66336827210750210261875024648

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