Properties

Label 8-896e4-1.1-c1e4-0-3
Degree $8$
Conductor $644513529856$
Sign $1$
Analytic cond. $2620.23$
Root an. cond. $2.67480$
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  − 4·7-s + 4·9-s − 8·17-s − 16·23-s + 12·25-s + 8·41-s + 32·47-s + 10·49-s − 16·63-s − 32·71-s + 24·73-s + 32·79-s + 2·81-s − 8·89-s − 40·97-s − 32·103-s + 24·113-s + 32·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·153-s + 157-s + ⋯
L(s)  = 1  − 1.51·7-s + 4/3·9-s − 1.94·17-s − 3.33·23-s + 12/5·25-s + 1.24·41-s + 4.66·47-s + 10/7·49-s − 2.01·63-s − 3.79·71-s + 2.80·73-s + 3.60·79-s + 2/9·81-s − 0.847·89-s − 4.06·97-s − 3.15·103-s + 2.25·113-s + 2.93·119-s + 1.09·121-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 − 2.58·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2620.23\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{896} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.007510473\)
\(L(\frac12)\) \(\approx\) \(1.007510473\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
good3$C_2^2:C_4$ \( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^2:C_4$ \( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 12 T^{2} - 18 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_4$ \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2:C_4$ \( 1 - 36 T^{2} + 1038 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 + 44 T^{2} + 2038 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2:C_4$ \( 1 - 116 T^{2} + 5974 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2:C_4$ \( 1 - 140 T^{2} + 8470 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
53$C_2^2:C_4$ \( 1 - 84 T^{2} + 5334 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 - 196 T^{2} + 16174 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 - 44 T^{2} + 5614 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 - 204 T^{2} + 18870 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2:C_4$ \( 1 - 132 T^{2} + 13134 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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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

−7.15124248992741200794844241104, −6.88485917240661119438985936688, −6.85433464557324392681858582816, −6.60956141677591059337507567762, −6.57541740603830524177403390152, −5.91178159885015951601624762899, −5.88253989446817293065956727907, −5.76694528978135966106523776545, −5.75409691354189271580749715436, −5.02969442222668144545760792905, −4.86674656817961731328386399614, −4.58121578400067069975973221691, −4.31907129274069995006206391116, −4.10580429523943148614150629175, −3.96903067483209967287134520277, −3.64335159958302093102705607241, −3.61024147735389999189844250735, −2.77590401298189243668772969266, −2.73586866346125114337555037798, −2.54655482498280317382189571805, −2.11700464203147875685397709880, −1.96274941220331534526711784378, −1.20216301084953963803582209224, −1.00603784400427532290366559574, −0.26990162487436964217273353815, 0.26990162487436964217273353815, 1.00603784400427532290366559574, 1.20216301084953963803582209224, 1.96274941220331534526711784378, 2.11700464203147875685397709880, 2.54655482498280317382189571805, 2.73586866346125114337555037798, 2.77590401298189243668772969266, 3.61024147735389999189844250735, 3.64335159958302093102705607241, 3.96903067483209967287134520277, 4.10580429523943148614150629175, 4.31907129274069995006206391116, 4.58121578400067069975973221691, 4.86674656817961731328386399614, 5.02969442222668144545760792905, 5.75409691354189271580749715436, 5.76694528978135966106523776545, 5.88253989446817293065956727907, 5.91178159885015951601624762899, 6.57541740603830524177403390152, 6.60956141677591059337507567762, 6.85433464557324392681858582816, 6.88485917240661119438985936688, 7.15124248992741200794844241104

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