Properties

Label 6-5290e3-1.1-c1e3-0-1
Degree $6$
Conductor $148035889000$
Sign $1$
Analytic cond. $75369.9$
Root an. cond. $6.49929$
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-s + 6·4-s + 3·5-s − 3·6-s + 3·7-s + 10·8-s − 9-s + 9·10-s + 5·11-s − 6·12-s − 3·13-s + 9·14-s − 3·15-s + 15·16-s + 13·17-s − 3·18-s − 5·19-s + 18·20-s − 3·21-s + 15·22-s − 10·24-s + 6·25-s − 9·26-s + 27-s + 18·28-s + 2·29-s + ⋯
L(s)  = 1  + 2.12·2-s − 0.577·3-s + 3·4-s + 1.34·5-s − 1.22·6-s + 1.13·7-s + 3.53·8-s − 1/3·9-s + 2.84·10-s + 1.50·11-s − 1.73·12-s − 0.832·13-s + 2.40·14-s − 0.774·15-s + 15/4·16-s + 3.15·17-s − 0.707·18-s − 1.14·19-s + 4.02·20-s − 0.654·21-s + 3.19·22-s − 2.04·24-s + 6/5·25-s − 1.76·26-s + 0.192·27-s + 3.40·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{3} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(75369.9\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5290} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{3} \cdot 23^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(37.52192116\)
\(L(\frac12)\) \(\approx\) \(37.52192116\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
23 \( 1 \)
good3$S_4\times C_2$ \( 1 + T + 2 T^{2} + 2 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 3 T + 6 T^{2} - 6 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 5 T + 34 T^{2} - 104 T^{3} + 34 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T + 24 T^{2} + 80 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 13 T + 100 T^{2} - 490 T^{3} + 100 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 5 T + 28 T^{2} + 4 p T^{3} + 28 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 7 T^{2} - 212 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 44 T^{2} - 386 T^{3} + 44 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 2 T - 5 T^{2} + 172 T^{3} - 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 5 T + 124 T^{2} - 404 T^{3} + 124 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 2 T + 49 T^{2} - 268 T^{3} + 49 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 25 T^{2} - 152 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 12 T + 135 T^{2} - 1200 T^{3} + 135 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 16 T + 181 T^{2} + 1456 T^{3} + 181 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 9 T + 42 T^{2} + 180 T^{3} + 42 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 8 T + 193 T^{2} + 1000 T^{3} + 193 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 13 T + 262 T^{2} + 1894 T^{3} + 262 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 22 T + 299 T^{2} + 2860 T^{3} + 299 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 4 T + 121 T^{2} - 104 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 8 T + 121 T^{2} + 1400 T^{3} + 121 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 6 T + 207 T^{2} - 780 T^{3} + 207 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 33 T + 588 T^{2} + 7050 T^{3} + 588 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.13773130339309901523791950570, −6.97977131620200437226204871285, −6.57978754610765752080528483882, −6.28644802762068845977856989467, −6.03351203531362294977258972798, −5.94390351060001613395692275553, −5.89887874168825603372319564766, −5.38028470484294048337054821729, −5.34857419105031557573951247275, −5.23779795035598359886424351412, −4.62660706438408970557836459821, −4.59986595275567226146711953372, −4.33838601913248892248371156637, −4.24628067195662812871512533452, −3.67165974937592120529633925396, −3.54044504110973782144627918519, −3.08833178563317813726101258943, −2.83511571947649564628885518628, −2.80554209144641365060493826734, −2.30597394252445569492147392538, −1.84390653122024934670579652928, −1.58324526046119323904355867773, −1.52263238377104808173335442210, −1.01241983022320852744024051929, −0.64186659045180251821653015316, 0.64186659045180251821653015316, 1.01241983022320852744024051929, 1.52263238377104808173335442210, 1.58324526046119323904355867773, 1.84390653122024934670579652928, 2.30597394252445569492147392538, 2.80554209144641365060493826734, 2.83511571947649564628885518628, 3.08833178563317813726101258943, 3.54044504110973782144627918519, 3.67165974937592120529633925396, 4.24628067195662812871512533452, 4.33838601913248892248371156637, 4.59986595275567226146711953372, 4.62660706438408970557836459821, 5.23779795035598359886424351412, 5.34857419105031557573951247275, 5.38028470484294048337054821729, 5.89887874168825603372319564766, 5.94390351060001613395692275553, 6.03351203531362294977258972798, 6.28644802762068845977856989467, 6.57978754610765752080528483882, 6.97977131620200437226204871285, 7.13773130339309901523791950570

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