Properties

Label 8-2352e4-1.1-c2e4-0-4
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
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  − 24·5-s − 6·9-s − 48·13-s − 24·17-s + 264·25-s + 96·29-s + 72·41-s + 144·45-s − 88·53-s − 96·61-s + 1.15e3·65-s − 48·73-s + 27·81-s + 576·85-s + 120·89-s − 432·97-s + 72·101-s − 48·109-s + 56·113-s + 288·117-s + 76·121-s − 1.46e3·125-s + 127-s + 131-s + 137-s + 139-s − 2.30e3·145-s + ⋯
L(s)  = 1  − 4.79·5-s − 2/3·9-s − 3.69·13-s − 1.41·17-s + 10.5·25-s + 3.31·29-s + 1.75·41-s + 16/5·45-s − 1.66·53-s − 1.57·61-s + 17.7·65-s − 0.657·73-s + 1/3·81-s + 6.77·85-s + 1.34·89-s − 4.45·97-s + 0.712·101-s − 0.440·109-s + 0.495·113-s + 2.46·117-s + 0.628·121-s − 11.7·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 15.8·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1413504263\)
\(L(\frac12)\) \(\approx\) \(0.1413504263\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( ( 1 + 12 T + 84 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 76 T^{2} - 10746 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + 24 T + 384 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 12 T + 564 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 506 T^{2} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 20 T^{2} + 186534 T^{4} + 20 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 48 T + 2186 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 140 T^{2} + 1478694 T^{4} + 140 p^{4} T^{6} + p^{8} T^{8} \)
37$C_2^2$ \( ( 1 + 1586 T^{2} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 36 T + 3588 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 1732 T^{2} + 3440358 T^{4} - 1732 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 - 8020 T^{2} + 25673574 T^{4} - 8020 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 44 T + 3510 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 5332 T^{2} + 13053126 T^{4} - 5332 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 48 T + 4656 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 16900 T^{2} + 111538854 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 11308 T^{2} + 69354150 T^{4} - 11308 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 24 T + 8352 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 16708 T^{2} + 131100678 T^{4} - 16708 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 23428 T^{2} + 227987238 T^{4} - 23428 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 - 60 T + 180 p T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 216 T + 30240 T^{2} + 216 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.29369475195929868833100784191, −5.88655428715285030252257031223, −5.81746045916065285190528142100, −5.60363968639035704168883596769, −5.07982843633272449540368974091, −4.79802682659762779531522824768, −4.77061235038182963897644852432, −4.69388079962380700039544758251, −4.48294631286003917358341532065, −4.30465017845430894763184470313, −4.26194134248766390492267804444, −3.74580631062668687812234778852, −3.73411003909499307587379950962, −3.33153689831589355583621366675, −3.28282983091396474947910600143, −2.75305071961008180343236585240, −2.68205966976484731410161240166, −2.58829430661141735741261377578, −2.45259108816678130963454721727, −1.70931202714474078684361452503, −1.59446125986311710729998264318, −0.807101003901574978472195774843, −0.57374048618133055322556145758, −0.37946949340947704206905592052, −0.14286424678677579197503860576, 0.14286424678677579197503860576, 0.37946949340947704206905592052, 0.57374048618133055322556145758, 0.807101003901574978472195774843, 1.59446125986311710729998264318, 1.70931202714474078684361452503, 2.45259108816678130963454721727, 2.58829430661141735741261377578, 2.68205966976484731410161240166, 2.75305071961008180343236585240, 3.28282983091396474947910600143, 3.33153689831589355583621366675, 3.73411003909499307587379950962, 3.74580631062668687812234778852, 4.26194134248766390492267804444, 4.30465017845430894763184470313, 4.48294631286003917358341532065, 4.69388079962380700039544758251, 4.77061235038182963897644852432, 4.79802682659762779531522824768, 5.07982843633272449540368974091, 5.60363968639035704168883596769, 5.81746045916065285190528142100, 5.88655428715285030252257031223, 6.29369475195929868833100784191

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