Properties

Label 8-2775e4-1.1-c0e4-0-4
Degree $8$
Conductor $5.930\times 10^{13}$
Sign $1$
Analytic cond. $3.67858$
Root an. cond. $1.17682$
Motivic weight $0$
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  + 2·4-s + 9-s + 16-s + 4·19-s − 4·31-s + 2·36-s − 49-s − 4·61-s − 2·64-s + 8·76-s − 2·79-s − 2·109-s + 4·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 4·171-s + 173-s + ⋯
L(s)  = 1  + 2·4-s + 9-s + 16-s + 4·19-s − 4·31-s + 2·36-s − 49-s − 4·61-s − 2·64-s + 8·76-s − 2·79-s − 2·109-s + 4·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 4·171-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(3.67858\)
Root analytic conductor: \(1.17682\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2775} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 37^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.815299436\)
\(L(\frac12)\) \(\approx\) \(2.815299436\)
\(L(1)\) 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)$
bad3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
7$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2$ \( ( 1 + T + T^{2} )^{4} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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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

−6.48798319002898440444146479794, −6.16760697365510773660104628599, −6.03535144523682538884445128663, −5.74020949622263983658758002605, −5.73798197336295451693772336730, −5.63707782312064713724533046933, −5.08224361122067076829324263509, −5.07261599624894089950998795339, −4.99893724517073529666204536598, −4.50758924548544409562797433188, −4.47510087751680836832660732501, −4.04966433195771582835647299868, −3.92876626880542804710367814715, −3.46283218285290825074213414428, −3.42844506330129041196371243280, −3.17714460936252941051868439030, −3.00507688105172873945131661509, −2.83792109104216925563622807242, −2.46580081951396923268192326839, −2.24627510075681699788047842975, −1.68563572102360427890900640987, −1.58968975608278260815546717874, −1.55249092406693099271471259034, −1.39691385612135329514453256174, −0.61142252351547762562900073435, 0.61142252351547762562900073435, 1.39691385612135329514453256174, 1.55249092406693099271471259034, 1.58968975608278260815546717874, 1.68563572102360427890900640987, 2.24627510075681699788047842975, 2.46580081951396923268192326839, 2.83792109104216925563622807242, 3.00507688105172873945131661509, 3.17714460936252941051868439030, 3.42844506330129041196371243280, 3.46283218285290825074213414428, 3.92876626880542804710367814715, 4.04966433195771582835647299868, 4.47510087751680836832660732501, 4.50758924548544409562797433188, 4.99893724517073529666204536598, 5.07261599624894089950998795339, 5.08224361122067076829324263509, 5.63707782312064713724533046933, 5.73798197336295451693772336730, 5.74020949622263983658758002605, 6.03535144523682538884445128663, 6.16760697365510773660104628599, 6.48798319002898440444146479794

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