Properties

Label 8-126e4-1.1-c2e4-0-6
Degree $8$
Conductor $252047376$
Sign $1$
Analytic cond. $138.938$
Root an. cond. $1.85290$
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  − 2·4-s + 14·7-s + 102·19-s − 26·25-s − 28·28-s − 42·31-s + 94·37-s + 124·43-s + 49·49-s − 288·61-s + 8·64-s + 62·67-s − 282·73-s − 204·76-s − 82·79-s + 52·100-s − 102·103-s − 338·109-s − 46·121-s + 84·124-s + 127-s + 131-s + 1.42e3·133-s + 137-s + 139-s − 188·148-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s + 2·7-s + 5.36·19-s − 1.03·25-s − 28-s − 1.35·31-s + 2.54·37-s + 2.88·43-s + 49-s − 4.72·61-s + 1/8·64-s + 0.925·67-s − 3.86·73-s − 2.68·76-s − 1.03·79-s + 0.519·100-s − 0.990·103-s − 3.10·109-s − 0.380·121-s + 0.677·124-s + 0.00787·127-s + 0.00763·131-s + 10.7·133-s + 0.00729·137-s + 0.00719·139-s − 1.27·148-s + 0.00671·149-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(138.938\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{126} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.312384895\)
\(L(\frac12)\) \(\approx\) \(3.312384895\)
\(L(2)\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 26 T^{2} + 51 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^3$ \( 1 + 46 T^{2} - 12525 T^{4} + 46 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2^2$ \( ( 1 - 335 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 554 T^{2} + 223395 T^{4} + 554 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 51 T + 1228 T^{2} - 51 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 986 T^{2} + 692355 T^{4} - 986 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 21 T + 1108 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 73 T + p^{2} T^{2} )^{2}( 1 + 26 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + 1342 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 31 T + p^{2} T^{2} )^{4} \)
47$C_2^3$ \( 1 - 2518 T^{2} + 1460643 T^{4} - 2518 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 214 T^{2} - 7844685 T^{4} + 214 p^{4} T^{6} + p^{8} T^{8} \)
59$C_2^3$ \( 1 + 26 T^{2} - 12116685 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8} \)
61$C_2^2$ \( ( 1 + 144 T + 10633 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 6554 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 141 T + 11956 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 41 T - 4560 T^{2} + 41 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 13754 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 12386 T^{2} + 90670755 T^{4} + 12386 p^{4} T^{6} + p^{8} T^{8} \)
97$C_2^2$ \( ( 1 - 17090 T^{2} + p^{4} T^{4} )^{2} \)
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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

−9.520453661739833870504139055403, −9.228615623349509868734665926540, −9.104983254988808411901091064404, −9.077109939226296806601124150937, −8.374140411513708548235747319416, −7.931782535984101302990678554889, −7.81328367083374943263496844457, −7.61111499328648994569052258525, −7.58828466632123859488811935787, −7.17300109856377330190283838977, −6.90634797381019122175765268389, −6.11240567253940469029673690585, −5.71278772400842597911187398348, −5.60307507354846873277134514169, −5.56683627772490433003093989572, −5.02746143245630392923879889175, −4.46962888631178437753923665646, −4.45354982415260915040560389120, −4.13883948716215062072365424699, −3.26997428528081853066328780507, −3.09189817563065502634708153849, −2.76280684712759249749233470435, −1.72949438704184250308697401804, −1.40164352030399668377685839969, −0.865737985357350770169734339365, 0.865737985357350770169734339365, 1.40164352030399668377685839969, 1.72949438704184250308697401804, 2.76280684712759249749233470435, 3.09189817563065502634708153849, 3.26997428528081853066328780507, 4.13883948716215062072365424699, 4.45354982415260915040560389120, 4.46962888631178437753923665646, 5.02746143245630392923879889175, 5.56683627772490433003093989572, 5.60307507354846873277134514169, 5.71278772400842597911187398348, 6.11240567253940469029673690585, 6.90634797381019122175765268389, 7.17300109856377330190283838977, 7.58828466632123859488811935787, 7.61111499328648994569052258525, 7.81328367083374943263496844457, 7.931782535984101302990678554889, 8.374140411513708548235747319416, 9.077109939226296806601124150937, 9.104983254988808411901091064404, 9.228615623349509868734665926540, 9.520453661739833870504139055403

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