Properties

Label 8-2352e4-1.1-c1e4-0-10
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
Motivic weight $1$
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  + 4·3-s + 6·9-s + 12·25-s − 4·27-s + 24·37-s − 24·47-s − 48·59-s + 48·75-s − 37·81-s − 24·83-s + 48·109-s + 96·111-s − 8·121-s + 127-s + 131-s + 137-s + 139-s − 96·141-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s − 192·177-s + 179-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 12/5·25-s − 0.769·27-s + 3.94·37-s − 3.50·47-s − 6.24·59-s + 5.54·75-s − 4.11·81-s − 2.63·83-s + 4.59·109-s + 9.11·111-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8.08·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s − 14.4·177-s + 0.0747·179-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.172688694\)
\(L(\frac12)\) \(\approx\) \(8.172688694\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + 120 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 48 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.28529455624340743276344681500, −6.26727172036099124958891431267, −5.99494124088692181499266466740, −5.86263887638049320262114602028, −5.76543016439949624170954802746, −5.11376802846146466904406921503, −5.07938130134522124117021659035, −4.65766525441782818240848838128, −4.56711300907880807425318834722, −4.55982760991144897946025918025, −4.44210978553710713871516798925, −3.87205174948274294178528533749, −3.54759432855569994749618497847, −3.53698701559598464905588566421, −3.27301797183950151530199763237, −2.96512269354220909518905118760, −2.71167593016259808640293614249, −2.70020294351996517244103150118, −2.69893896701501539368299981896, −2.06344160931186880622935419822, −1.65059741042856963922540564516, −1.55372952903544133833840960395, −1.46324327517915080267546423612, −0.71922263061413141927764647362, −0.39297994716565131233365283295, 0.39297994716565131233365283295, 0.71922263061413141927764647362, 1.46324327517915080267546423612, 1.55372952903544133833840960395, 1.65059741042856963922540564516, 2.06344160931186880622935419822, 2.69893896701501539368299981896, 2.70020294351996517244103150118, 2.71167593016259808640293614249, 2.96512269354220909518905118760, 3.27301797183950151530199763237, 3.53698701559598464905588566421, 3.54759432855569994749618497847, 3.87205174948274294178528533749, 4.44210978553710713871516798925, 4.55982760991144897946025918025, 4.56711300907880807425318834722, 4.65766525441782818240848838128, 5.07938130134522124117021659035, 5.11376802846146466904406921503, 5.76543016439949624170954802746, 5.86263887638049320262114602028, 5.99494124088692181499266466740, 6.26727172036099124958891431267, 6.28529455624340743276344681500

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