Properties

Label 6-5290e3-1.1-c1e3-0-0
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 − 3·11-s + 6·12-s − 13-s − 9·14-s + 3·15-s + 15·16-s + 7·17-s + 3·18-s − 3·19-s + 18·20-s − 3·21-s − 9·22-s + 10·24-s + 6·25-s − 3·26-s − 5·27-s − 18·28-s − 4·29-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 − 0.904·11-s + 1.73·12-s − 0.277·13-s − 2.40·14-s + 0.774·15-s + 15/4·16-s + 1.69·17-s + 0.707·18-s − 0.688·19-s + 4.02·20-s − 0.654·21-s − 1.91·22-s + 2.04·24-s + 6/5·25-s − 0.588·26-s − 0.962·27-s − 3.40·28-s − 0.742·29-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

\(L(1)\) \(\approx\) \(12.78153702\)
\(L(\frac12)\) \(\approx\) \(12.78153702\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 - T + 2 p T^{3} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 3 T - 22 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T - 6 T^{2} - 78 T^{3} - 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + T + 24 T^{2} + 8 T^{3} + 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 7 T + 58 T^{2} - 220 T^{3} + 58 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 36 T^{2} + 50 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 55 T^{2} + 256 T^{3} + 55 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 5 T + 86 T^{2} + 302 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 2 T + 71 T^{2} - 116 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - T + 64 T^{2} + 104 T^{3} + 64 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{3} \)
47$S_4\times C_2$ \( 1 + 14 T + 145 T^{2} + 1028 T^{3} + 145 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 - 14 T + 205 T^{2} - 1508 T^{3} + 205 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + T + 26 T^{2} - 404 T^{3} + 26 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
67$D_{6}$ \( 1 + 8 T + 57 T^{2} + 688 T^{3} + 57 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 11 T + 244 T^{2} - 1586 T^{3} + 244 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 8 T + 179 T^{2} + 920 T^{3} + 179 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 4 T - 3 T^{2} + 520 T^{3} - 3 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 8 T + 229 T^{2} + 1232 T^{3} + 229 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 18 T + 219 T^{2} + 2052 T^{3} + 219 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 33 T + 570 T^{2} - 6568 T^{3} + 570 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.13109420931190164676078627216, −7.05800244529518647377369516779, −6.60186370564124759268677301951, −6.48352626563470051816625721910, −6.06684882040549659096926270195, −6.03222030876671724833400775412, −5.69566366078123926162477055323, −5.51032919257011054491803318402, −5.31998746870685384338984166895, −5.10332184670167621500059390160, −4.83721550610330969470009775061, −4.66610523362160838657701776569, −4.04559357599305318209509754599, −3.91010174237304585126725868440, −3.76286995378573907334876479545, −3.44751216237722378240481449055, −3.04415772202183749083416394996, −2.92297660399449295998224733165, −2.88157900574706641779196063538, −2.12394577228632665167922097117, −2.06754207227145485152371487459, −1.97621079577623223500824879106, −1.43852526136901948256193569870, −1.07533825837525047937364396011, −0.30239563090186820739142845946, 0.30239563090186820739142845946, 1.07533825837525047937364396011, 1.43852526136901948256193569870, 1.97621079577623223500824879106, 2.06754207227145485152371487459, 2.12394577228632665167922097117, 2.88157900574706641779196063538, 2.92297660399449295998224733165, 3.04415772202183749083416394996, 3.44751216237722378240481449055, 3.76286995378573907334876479545, 3.91010174237304585126725868440, 4.04559357599305318209509754599, 4.66610523362160838657701776569, 4.83721550610330969470009775061, 5.10332184670167621500059390160, 5.31998746870685384338984166895, 5.51032919257011054491803318402, 5.69566366078123926162477055323, 6.03222030876671724833400775412, 6.06684882040549659096926270195, 6.48352626563470051816625721910, 6.60186370564124759268677301951, 7.05800244529518647377369516779, 7.13109420931190164676078627216

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