Properties

Label 6-7098e3-1.1-c1e3-0-2
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 − 9·6-s − 3·7-s + 10·8-s + 6·9-s + 11-s − 18·12-s − 9·14-s + 15·16-s + 9·17-s + 18·18-s + 19-s + 9·21-s + 3·22-s + 23-s − 30·24-s − 8·25-s − 10·27-s − 18·28-s − 2·29-s + 4·31-s + 21·32-s − 3·33-s + 27·34-s + 36·36-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s + 0.301·11-s − 5.19·12-s − 2.40·14-s + 15/4·16-s + 2.18·17-s + 4.24·18-s + 0.229·19-s + 1.96·21-s + 0.639·22-s + 0.208·23-s − 6.12·24-s − 8/5·25-s − 1.92·27-s − 3.40·28-s − 0.371·29-s + 0.718·31-s + 3.71·32-s − 0.522·33-s + 4.63·34-s + 6·36-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

\(L(1)\) \(\approx\) \(13.36478101\)
\(L(\frac12)\) \(\approx\) \(13.36478101\)
\(L(\frac{3}{2})\) 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$A_4\times C_2$ \( 1 + 8 T^{2} + 7 T^{3} + 8 p T^{4} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - T + 17 T^{2} + 7 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
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$A_4\times C_2$ \( 1 - T + 41 T^{2} - 9 T^{3} + 41 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - T + 25 T^{2} - 129 T^{3} + 25 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 2 T + 2 p T^{2} + 45 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 4 T + 54 T^{2} - 79 T^{3} + 54 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 14 T + 146 T^{2} - 1029 T^{3} + 146 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 6 T + 86 T^{2} - 11 p T^{3} + 86 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 29 T + 407 T^{2} + 3375 T^{3} + 407 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 7 T + 120 T^{2} - 567 T^{3} + 120 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 16 T + 214 T^{2} - 1653 T^{3} + 214 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 10 T + 208 T^{2} + 1209 T^{3} + 208 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 11 T + 186 T^{2} - 1271 T^{3} + 186 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 2 T + 116 T^{2} - 519 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 14 T + 241 T^{2} - 1932 T^{3} + 241 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 7 T + 9 T^{2} + 245 T^{3} + 9 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 2 T + 26 T^{2} - 945 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 5 T + 171 T^{2} + 411 T^{3} + 171 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 120 T^{2} - 497 T^{3} + 120 p T^{4} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 6 T + 86 T^{2} - 1921 T^{3} + 86 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.97571308322249683287413505120, −6.67737889969777734493315144302, −6.31114603329833797337362154568, −6.20834011992144226178335123324, −5.91868047272880425165128077794, −5.77927732917681737026692716288, −5.73214210635757272560599836299, −5.27288193137734556227974655274, −5.16264770486478922947440877943, −5.08897951785586896716068707039, −4.49454858810118512447501815079, −4.47625231756729024719877778905, −4.15662587948371313749634278917, −3.83757511907556622262080628487, −3.66938354277451455100568008582, −3.48617716682372907270574098783, −3.03652081732347157195545113311, −2.99267196603154114743720235527, −2.63412286386940625106858155014, −2.08178636012757952688851218408, −1.82253461326485157326253031006, −1.69405882352552328190478774719, −0.990368982624148812032734833212, −0.70419552267290399117817170350, −0.56881597555184533844271705534, 0.56881597555184533844271705534, 0.70419552267290399117817170350, 0.990368982624148812032734833212, 1.69405882352552328190478774719, 1.82253461326485157326253031006, 2.08178636012757952688851218408, 2.63412286386940625106858155014, 2.99267196603154114743720235527, 3.03652081732347157195545113311, 3.48617716682372907270574098783, 3.66938354277451455100568008582, 3.83757511907556622262080628487, 4.15662587948371313749634278917, 4.47625231756729024719877778905, 4.49454858810118512447501815079, 5.08897951785586896716068707039, 5.16264770486478922947440877943, 5.27288193137734556227974655274, 5.73214210635757272560599836299, 5.77927732917681737026692716288, 5.91868047272880425165128077794, 6.20834011992144226178335123324, 6.31114603329833797337362154568, 6.67737889969777734493315144302, 6.97571308322249683287413505120

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