Properties

Label 8-2880e4-1.1-c2e4-0-9
Degree $8$
Conductor $6.880\times 10^{13}$
Sign $1$
Analytic cond. $3.79236\times 10^{7}$
Root an. cond. $8.85857$
Motivic weight $2$
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  − 16·7-s − 8·13-s − 32·19-s − 10·25-s + 8·31-s + 136·37-s − 80·43-s + 144·49-s + 40·61-s + 304·67-s + 152·73-s + 200·79-s + 128·91-s − 424·97-s − 112·103-s − 104·109-s + 88·121-s + 127-s + 131-s + 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2.28·7-s − 0.615·13-s − 1.68·19-s − 2/5·25-s + 8/31·31-s + 3.67·37-s − 1.86·43-s + 2.93·49-s + 0.655·61-s + 4.53·67-s + 2.08·73-s + 2.53·79-s + 1.40·91-s − 4.37·97-s − 1.08·103-s − 0.954·109-s + 8/11·121-s + 0.00787·127-s + 0.00763·131-s + 3.84·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3.79236\times 10^{7}\)
Root analytic conductor: \(8.85857\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2880} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7531124876\)
\(L(\frac12)\) \(\approx\) \(0.7531124876\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
good7$D_{4}$ \( ( 1 + 8 T + 24 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + 4 T + 252 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 16 T + 426 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 1612 T^{2} + 1157478 T^{4} - 1612 p^{4} T^{6} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 - 4 T + 1566 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 68 T + 3084 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 40 T + 3738 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 340 T^{2} - 372378 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 20 T + 4302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 76 T + p^{2} T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 76 T + 8862 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 100 T + 11742 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 21652 T^{2} + 211289478 T^{4} - 21652 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 + 212 T + 29694 T^{2} + 212 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.21881703075578863977763578287, −5.84244726416050522283063949889, −5.68204346820005332631690490509, −5.31458991174551035590850879045, −5.25080577518498545709925877548, −5.20690837318545624164673372360, −4.71662452877760202553494275009, −4.62020967541014593362133648943, −4.17974991680375483144716163003, −4.17916768644048147655878025408, −3.86216652590826416103297660016, −3.79802390209566812433652428655, −3.45366884666049976908480489175, −3.40813934341592492468609471910, −2.91383745913271619718211748796, −2.67861688137078239359607768648, −2.52074632149890820205208226825, −2.51666465656612974725774855539, −2.05427626253472925262526158507, −1.93544225375136720590782335511, −1.47579951102444598491255350780, −0.882258569866188961204681124014, −0.866539520986737301996674551145, −0.49749027980153699482417549770, −0.13796071161197140231592230668, 0.13796071161197140231592230668, 0.49749027980153699482417549770, 0.866539520986737301996674551145, 0.882258569866188961204681124014, 1.47579951102444598491255350780, 1.93544225375136720590782335511, 2.05427626253472925262526158507, 2.51666465656612974725774855539, 2.52074632149890820205208226825, 2.67861688137078239359607768648, 2.91383745913271619718211748796, 3.40813934341592492468609471910, 3.45366884666049976908480489175, 3.79802390209566812433652428655, 3.86216652590826416103297660016, 4.17916768644048147655878025408, 4.17974991680375483144716163003, 4.62020967541014593362133648943, 4.71662452877760202553494275009, 5.20690837318545624164673372360, 5.25080577518498545709925877548, 5.31458991174551035590850879045, 5.68204346820005332631690490509, 5.84244726416050522283063949889, 6.21881703075578863977763578287

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