Properties

Label 6-8550e3-1.1-c1e3-0-13
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 $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s − 10·8-s + 4·11-s + 6·13-s + 15·16-s − 10·17-s − 3·19-s − 12·22-s − 6·23-s − 18·26-s − 10·29-s + 14·31-s − 21·32-s + 30·34-s + 2·37-s + 9·38-s − 10·43-s + 24·44-s + 18·46-s + 2·47-s − 5·49-s + 36·52-s − 14·53-s + 30·58-s − 14·59-s + 6·61-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s − 3.53·8-s + 1.20·11-s + 1.66·13-s + 15/4·16-s − 2.42·17-s − 0.688·19-s − 2.55·22-s − 1.25·23-s − 3.53·26-s − 1.85·29-s + 2.51·31-s − 3.71·32-s + 5.14·34-s + 0.328·37-s + 1.45·38-s − 1.52·43-s + 3.61·44-s + 2.65·46-s + 0.291·47-s − 5/7·49-s + 4.99·52-s − 1.92·53-s + 3.93·58-s − 1.82·59-s + 0.768·61-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: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
3 \( 1 \)
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$S_4\times C_2$ \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 10 T + 71 T^{2} + 332 T^{3} + 71 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 65 T^{2} + 268 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 14 T + 121 T^{2} - 716 T^{3} + 121 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 2 T + 27 T^{2} - 380 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 107 T^{2} + 16 T^{3} + 107 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 10 T + 125 T^{2} + 852 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 2 T + 89 T^{2} - 4 T^{3} + 89 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 14 T + 171 T^{2} + 1332 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 14 T + 229 T^{2} + 1692 T^{3} + 229 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 133 T^{2} - 504 T^{3} + 133 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 - 22 T + 361 T^{2} - 3676 T^{3} + 361 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 129 T^{2} - 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 4 T + 219 T^{2} + 792 T^{3} + 219 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 6 T + 111 T^{2} + 948 T^{3} + 111 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.34259352230170570617543730828, −6.80112782038922583156676780769, −6.76940276498606673949460970540, −6.60877166144834602510749654782, −6.28695242462468738448390702846, −6.26375615380323349361466640135, −6.21470719207236599319447571225, −5.90933538391036397910953443821, −5.33155595652291036965448604762, −5.30722789537876786333152559870, −4.80524783243909063814349744353, −4.59799295266907272373988751705, −4.30933140601898424930468949964, −4.02136733261253696001527034117, −3.76343431252083552332182407765, −3.73965198960801103894794851466, −3.19552415553085885307725142898, −2.88387963619577963175135864524, −2.79246547229151501379194570727, −2.13325000183266475115800512502, −2.01682000908055418583122834502, −2.01454928092403901511776494726, −1.31568274689221893632334716861, −1.17736573232822226794240939826, −1.14527809534458174151834125722, 0, 0, 0, 1.14527809534458174151834125722, 1.17736573232822226794240939826, 1.31568274689221893632334716861, 2.01454928092403901511776494726, 2.01682000908055418583122834502, 2.13325000183266475115800512502, 2.79246547229151501379194570727, 2.88387963619577963175135864524, 3.19552415553085885307725142898, 3.73965198960801103894794851466, 3.76343431252083552332182407765, 4.02136733261253696001527034117, 4.30933140601898424930468949964, 4.59799295266907272373988751705, 4.80524783243909063814349744353, 5.30722789537876786333152559870, 5.33155595652291036965448604762, 5.90933538391036397910953443821, 6.21470719207236599319447571225, 6.26375615380323349361466640135, 6.28695242462468738448390702846, 6.60877166144834602510749654782, 6.76940276498606673949460970540, 6.80112782038922583156676780769, 7.34259352230170570617543730828

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