Properties

Degree $8$
Conductor $9.041\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 8·13-s + 3·16-s − 16·52-s − 4·64-s − 48·67-s − 81-s + 24·89-s − 40·101-s − 32·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4-s + 2.21·13-s + 3/4·16-s − 2.21·52-s − 1/2·64-s − 5.86·67-s − 1/9·81-s + 2.54·89-s − 3.98·101-s − 3.15·103-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 17^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1734} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 17^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.123151129\)
\(L(\frac12)\) \(\approx\) \(2.123151129\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{4} \)
17 \( 1 \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 206 T^{4} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
29$C_2^2$$\times$$C_2^2$ \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \)
31$C_2^3$ \( 1 - 1918 T^{4} + p^{4} T^{8} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 24 T^{2} + p^{2} T^{4} )( 1 + 24 T^{2} + p^{2} T^{4} ) \)
41$C_2^2$$\times$$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} ) \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$$\times$$C_2^2$ \( ( 1 - 120 T^{2} + p^{2} T^{4} )( 1 + 120 T^{2} + p^{2} T^{4} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 - 8542 T^{4} + p^{4} T^{8} \)
79$C_2^3$ \( 1 - 3646 T^{4} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
97$C_2^3$ \( 1 - 18814 T^{4} + p^{4} T^{8} \)
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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

−6.73762330629450599688885758334, −6.24323155993572064555889166174, −6.22999711527187383113308793123, −5.88669135276707711730224983634, −5.80467154241489835970827918340, −5.64567711561840772459996967512, −5.46395910157928627437454571796, −5.00037074411665176561237703695, −4.98838794897735425210161886000, −4.52141574697682164516372558356, −4.40296691061677148069820182139, −4.31154389756146238155898685732, −3.90869108617668724053920777837, −3.89336177111704867076803382650, −3.47989323980037405429060564971, −3.25782635837158200272772839711, −3.07830398946111315135958691862, −2.71123168396102281928516274806, −2.66313317371066589482155853989, −2.02802793126179874867724678641, −1.60428613811992023518741227950, −1.47636988186842666570485247348, −1.32624683027231639855781763160, −0.69169905610264995419003434904, −0.34428919154455157982916624769, 0.34428919154455157982916624769, 0.69169905610264995419003434904, 1.32624683027231639855781763160, 1.47636988186842666570485247348, 1.60428613811992023518741227950, 2.02802793126179874867724678641, 2.66313317371066589482155853989, 2.71123168396102281928516274806, 3.07830398946111315135958691862, 3.25782635837158200272772839711, 3.47989323980037405429060564971, 3.89336177111704867076803382650, 3.90869108617668724053920777837, 4.31154389756146238155898685732, 4.40296691061677148069820182139, 4.52141574697682164516372558356, 4.98838794897735425210161886000, 5.00037074411665176561237703695, 5.46395910157928627437454571796, 5.64567711561840772459996967512, 5.80467154241489835970827918340, 5.88669135276707711730224983634, 6.22999711527187383113308793123, 6.24323155993572064555889166174, 6.73762330629450599688885758334

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