Properties

Label 8-462e4-1.1-c1e4-0-8
Degree $8$
Conductor $45558341136$
Sign $1$
Analytic cond. $185.215$
Root an. cond. $1.92070$
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  − 2·3-s − 2·4-s − 4·5-s + 6·7-s + 2·9-s + 4·12-s + 8·15-s + 3·16-s + 20·17-s + 8·20-s − 12·21-s − 6·27-s − 12·28-s − 24·35-s − 4·36-s + 4·41-s + 8·43-s − 8·45-s + 24·47-s − 6·48-s + 18·49-s − 40·51-s + 4·59-s − 16·60-s + 12·63-s − 4·64-s + 24·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.78·5-s + 2.26·7-s + 2/3·9-s + 1.15·12-s + 2.06·15-s + 3/4·16-s + 4.85·17-s + 1.78·20-s − 2.61·21-s − 1.15·27-s − 2.26·28-s − 4.05·35-s − 2/3·36-s + 0.624·41-s + 1.21·43-s − 1.19·45-s + 3.50·47-s − 0.866·48-s + 18/7·49-s − 5.60·51-s + 0.520·59-s − 2.06·60-s + 1.51·63-s − 1/2·64-s + 2.93·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(185.215\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{462} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.474441103\)
\(L(\frac12)\) \(\approx\) \(1.474441103\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
17$D_{4}$ \( ( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \)
29$C_4\times C_2$ \( 1 - 44 T^{2} + 886 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 120 T^{2} + 9038 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_4\times C_2$ \( 1 - 76 T^{2} + 3766 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 16 T^{2} - 2354 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 24 T^{2} + 6302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
89$D_{4}$ \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 340 T^{2} + 47398 T^{4} - 340 p^{2} T^{6} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.960195100297008361430514007303, −7.946442717532163595655263088628, −7.50927110551669740277263008710, −7.45913893790855839235613836933, −7.36203227983659590319131766139, −6.77201429867421611585224717694, −6.64786886128596789253957281411, −5.85051485121645107465174064908, −5.82434976170426261929661264938, −5.63996305178035765135593991040, −5.58059461009239174623110925924, −5.12833905337014648757307953939, −5.11231936387832528042773213988, −4.67859075981237580197284009778, −4.41055849278307176460139108080, −3.97153044460235755914528851188, −3.80645174564738535204300583265, −3.79730004063956671084240800959, −3.43125204184876553210421189297, −2.96460290260980045292561963437, −2.31279278911285762268133530575, −2.02024850973126980547445189196, −1.17292470199172224556447731944, −1.01607175494390905174026577253, −0.69097073518968889309072920179, 0.69097073518968889309072920179, 1.01607175494390905174026577253, 1.17292470199172224556447731944, 2.02024850973126980547445189196, 2.31279278911285762268133530575, 2.96460290260980045292561963437, 3.43125204184876553210421189297, 3.79730004063956671084240800959, 3.80645174564738535204300583265, 3.97153044460235755914528851188, 4.41055849278307176460139108080, 4.67859075981237580197284009778, 5.11231936387832528042773213988, 5.12833905337014648757307953939, 5.58059461009239174623110925924, 5.63996305178035765135593991040, 5.82434976170426261929661264938, 5.85051485121645107465174064908, 6.64786886128596789253957281411, 6.77201429867421611585224717694, 7.36203227983659590319131766139, 7.45913893790855839235613836933, 7.50927110551669740277263008710, 7.946442717532163595655263088628, 7.960195100297008361430514007303

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