Properties

Label 6-8550e3-1.1-c1e3-0-5
Degree $6$
Conductor $625026375000$
Sign $1$
Analytic cond. $318221.$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s − 2·7-s − 10·8-s + 5·11-s + 6·14-s + 15·16-s + 6·17-s + 3·19-s − 15·22-s − 23-s − 12·28-s + 15·29-s − 31-s − 21·32-s − 18·34-s + 4·37-s − 9·38-s + 6·41-s − 10·43-s + 30·44-s + 3·46-s − 7·49-s − 5·53-s + 20·56-s − 45·58-s + 12·59-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s − 0.755·7-s − 3.53·8-s + 1.50·11-s + 1.60·14-s + 15/4·16-s + 1.45·17-s + 0.688·19-s − 3.19·22-s − 0.208·23-s − 2.26·28-s + 2.78·29-s − 0.179·31-s − 3.71·32-s − 3.08·34-s + 0.657·37-s − 1.45·38-s + 0.937·41-s − 1.52·43-s + 4.52·44-s + 0.442·46-s − 49-s − 0.686·53-s + 2.67·56-s − 5.90·58-s + 1.56·59-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(318221.\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8550} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.534961442\)
\(L(\frac12)\) \(\approx\) \(2.534961442\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + 2 T + 11 T^{2} + 26 T^{3} + 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 5 T + 36 T^{2} - 107 T^{3} + 36 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T^{2} - 82 T^{3} + 3 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 6 T + 45 T^{2} - 150 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + T + 60 T^{2} + 49 T^{3} + 60 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 15 T + 144 T^{2} - 879 T^{3} + 144 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + T + 8 T^{2} - 47 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 95 T^{2} - 280 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 51 T^{2} - 168 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 10 T + 157 T^{2} + 878 T^{3} + 157 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 117 T^{2} + 36 T^{3} + 117 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 5 T + 102 T^{2} + 347 T^{3} + 102 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T - 27 T^{2} + 1050 T^{3} - 27 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 46 T^{2} + 517 T^{3} + 46 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + T + 154 T^{2} - 7 T^{3} + 154 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 24 T + 387 T^{2} - 3750 T^{3} + 387 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 9 T + 138 T^{2} + 773 T^{3} + 138 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 5 T + 224 T^{2} - 749 T^{3} + 224 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 11 T + 222 T^{2} + 1787 T^{3} + 222 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 23 T + 234 T^{2} - 1787 T^{3} + 234 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 14 T + 197 T^{2} + 1742 T^{3} + 197 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.98391516173195293346929411688, −6.70822152820680212178004612805, −6.48583285273010309281117234839, −6.38856604352860129650521657557, −6.08598786205788596026646160184, −5.90898931372969171529282408886, −5.70268547511444385925728157917, −5.20912904338844590485908789938, −5.13156305196487188553955169591, −4.82690149840755040528320987641, −4.46742168215225552774174233820, −4.23642982897248093565322646159, −3.79974461881919149099740172065, −3.53871729033382836877546586842, −3.33223892638905564821696806457, −3.29961156470174877749186120321, −2.72079310955885033452393550227, −2.53306045917473746525591495939, −2.48269933570261974655670865807, −1.74894419392809213818377107149, −1.51653150576357344621389364664, −1.47553248552986625401534371939, −0.837775023065338820238517567672, −0.67516742979317141808597963033, −0.51874825983658149942052713891, 0.51874825983658149942052713891, 0.67516742979317141808597963033, 0.837775023065338820238517567672, 1.47553248552986625401534371939, 1.51653150576357344621389364664, 1.74894419392809213818377107149, 2.48269933570261974655670865807, 2.53306045917473746525591495939, 2.72079310955885033452393550227, 3.29961156470174877749186120321, 3.33223892638905564821696806457, 3.53871729033382836877546586842, 3.79974461881919149099740172065, 4.23642982897248093565322646159, 4.46742168215225552774174233820, 4.82690149840755040528320987641, 5.13156305196487188553955169591, 5.20912904338844590485908789938, 5.70268547511444385925728157917, 5.90898931372969171529282408886, 6.08598786205788596026646160184, 6.38856604352860129650521657557, 6.48583285273010309281117234839, 6.70822152820680212178004612805, 6.98391516173195293346929411688

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