Properties

Label 6-7098e3-1.1-c1e3-0-3
Degree $6$
Conductor $357608625192$
Sign $1$
Analytic cond. $182070.$
Root an. cond. $7.52846$
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  + 3·2-s − 3·3-s + 6·4-s + 3·5-s − 9·6-s − 3·7-s + 10·8-s + 6·9-s + 9·10-s − 3·11-s − 18·12-s − 9·14-s − 9·15-s + 15·16-s − 9·17-s + 18·18-s + 11·19-s + 18·20-s + 9·21-s − 9·22-s + 11·23-s − 30·24-s − 2·25-s − 10·27-s − 18·28-s + 8·29-s − 27·30-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s − 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s + 2.84·10-s − 0.904·11-s − 5.19·12-s − 2.40·14-s − 2.32·15-s + 15/4·16-s − 2.18·17-s + 4.24·18-s + 2.52·19-s + 4.02·20-s + 1.96·21-s − 1.91·22-s + 2.29·23-s − 6.12·24-s − 2/5·25-s − 1.92·27-s − 3.40·28-s + 1.48·29-s − 4.92·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(182070.\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(17.63277895\)
\(L(\frac12)\) \(\approx\) \(17.63277895\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good5$A_4\times C_2$ \( 1 - 3 T + 11 T^{2} - 31 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 3 T + 15 T^{2} + 53 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 9 T + 57 T^{2} + 277 T^{3} + 57 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 11 T + 53 T^{2} - 207 T^{3} + 53 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 11 T + 93 T^{2} - 519 T^{3} + 93 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 8 T + 71 T^{2} - 400 T^{3} + 71 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 11 T + 89 T^{2} - 471 T^{3} + 89 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 3 T - 19 T^{2} + 211 T^{3} - 19 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 21 T + 249 T^{2} + 1925 T^{3} + 249 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 10 T + 97 T^{2} - 532 T^{3} + 97 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 14 T + 3 p T^{2} + 924 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 2 T + 39 T^{2} - 540 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 32 T + 509 T^{2} - 4888 T^{3} + 509 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 6 T - T^{2} - 380 T^{3} - p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 10 T + 197 T^{2} - 1236 T^{3} + 197 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 21 T + 143 T^{2} + 623 T^{3} + 143 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 16 T + 239 T^{2} - 2328 T^{3} + 239 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 4 T + 233 T^{2} - 624 T^{3} + 233 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 22 T + 401 T^{2} - 3980 T^{3} + 401 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 21 T + 365 T^{2} + 3689 T^{3} + 365 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 36 T + 695 T^{2} - 8432 T^{3} + 695 p T^{4} - 36 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.86119405662943590164849080497, −6.62953736193368687120491264964, −6.46625138155622114803946481056, −6.26703008594601874714569004943, −6.00681172260235408924471831463, −5.75933181280254690549751143396, −5.58385911914567857426757897958, −5.29897141208485115496287136129, −5.09327453961208513195758704707, −4.89399667989101601478955276663, −4.79789658553313099990757923798, −4.58592351449154998824457586394, −4.34037511374581379055308351965, −3.74823239854978390678139938892, −3.63678391127054125405137949878, −3.36922615686282332945954399746, −2.98872046369441584219763531797, −2.74315647899029034482060005184, −2.73959738439111285985215294054, −2.01451710525541718874552850798, −1.98225076921600848391361714243, −1.76850083322328129889015015279, −0.907321693321209489941544229100, −0.808399871927129923466486978631, −0.59910144327633140994847215715, 0.59910144327633140994847215715, 0.808399871927129923466486978631, 0.907321693321209489941544229100, 1.76850083322328129889015015279, 1.98225076921600848391361714243, 2.01451710525541718874552850798, 2.73959738439111285985215294054, 2.74315647899029034482060005184, 2.98872046369441584219763531797, 3.36922615686282332945954399746, 3.63678391127054125405137949878, 3.74823239854978390678139938892, 4.34037511374581379055308351965, 4.58592351449154998824457586394, 4.79789658553313099990757923798, 4.89399667989101601478955276663, 5.09327453961208513195758704707, 5.29897141208485115496287136129, 5.58385911914567857426757897958, 5.75933181280254690549751143396, 6.00681172260235408924471831463, 6.26703008594601874714569004943, 6.46625138155622114803946481056, 6.62953736193368687120491264964, 6.86119405662943590164849080497

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