Properties

Label 8-13e8-1.1-c3e4-0-1
Degree $8$
Conductor $815730721$
Sign $1$
Analytic cond. $9885.78$
Root an. cond. $3.15774$
Motivic weight $3$
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 + 58·9-s + 64·16-s − 26·17-s + 156·23-s − 78·25-s − 416·27-s − 394·29-s − 312·43-s − 256·48-s − 286·49-s + 104·51-s + 372·53-s − 290·61-s − 624·69-s + 312·75-s − 304·79-s + 1.96e3·81-s + 1.57e3·87-s − 1.63e3·101-s − 6.55e3·103-s − 1.04e3·107-s − 654·113-s − 1.63e3·121-s + 127-s + 1.24e3·129-s + 131-s + ⋯
L(s)  = 1  − 0.769·3-s + 2.14·9-s + 16-s − 0.370·17-s + 1.41·23-s − 0.623·25-s − 2.96·27-s − 2.52·29-s − 1.10·43-s − 0.769·48-s − 0.833·49-s + 0.285·51-s + 0.964·53-s − 0.608·61-s − 1.08·69-s + 0.480·75-s − 0.432·79-s + 2.68·81-s + 1.94·87-s − 1.61·101-s − 6.26·103-s − 0.943·107-s − 0.544·113-s − 1.23·121-s + 0.000698·127-s + 0.851·129-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(13^{8}\)
Sign: $1$
Analytic conductor: \(9885.78\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{169} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.2694747472\)
\(L(\frac12)\) \(\approx\) \(0.2694747472\)
\(L(\frac{5}{2})\) 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)$
bad13 \( 1 \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} )( 1 + p^{2} T + p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} ) \)
3$C_2^2$ \( ( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 39 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 397 T^{2} + p^{6} T^{4} )( 1 + 683 T^{2} + p^{6} T^{4} ) \)
11$C_2^3$ \( 1 + 1638 T^{2} + 911483 T^{4} + 1638 p^{6} T^{6} + p^{12} T^{8} \)
17$C_2^2$ \( ( 1 + 13 T - 4744 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$C_2^3$ \( 1 + 12818 T^{2} + 117255243 T^{4} + 12818 p^{6} T^{6} + p^{12} T^{8} \)
23$C_2^2$ \( ( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 197 T + 14420 T^{2} + 197 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 54106 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 49777 T^{2} - 87976680 T^{4} + 49777 p^{6} T^{6} + p^{12} T^{8} \)
41$C_2^3$ \( 1 + 110617 T^{2} + 7486016448 T^{4} + 110617 p^{6} T^{6} + p^{12} T^{8} \)
43$C_2^2$ \( ( 1 + 156 T - 55171 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 181402 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 93 T + p^{3} T^{2} )^{4} \)
59$C_2^3$ \( 1 - 335738 T^{2} + 70539471003 T^{4} - 335738 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2^2$ \( ( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 141518 T^{2} - 70431037845 T^{4} - 141518 p^{6} T^{6} + p^{12} T^{8} \)
71$C_2^3$ \( 1 + 288106 T^{2} - 45095216685 T^{4} + 288106 p^{6} T^{6} + p^{12} T^{8} \)
73$C_2^2$ \( ( 1 - 731809 T^{2} + p^{6} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 76 T + p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 749190 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 1339182 T^{2} + 1296427138163 T^{4} + 1339182 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2^3$ \( 1 + 1768702 T^{2} + 2295334759875 T^{4} + 1768702 p^{6} T^{6} + p^{12} T^{8} \)
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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

−9.147633935025717020481531939898, −8.726663677637035073899286257173, −8.016155787730277047372686460105, −7.933881390920548303463260050649, −7.84204876759055956765639938549, −7.65691460233254945002174343382, −6.93545204156817504678880127090, −6.83875942143023981378150222726, −6.80222687969741269464423094826, −6.60916279389742401079830992922, −5.79331407752267733878280277254, −5.56497464752801068374341290045, −5.42685636239814820828882099831, −5.40454543706290499328685267004, −4.71314616758000778788612895226, −4.23142765820583406878319203939, −4.11360195887226238013533721709, −3.96009558417502354075370650205, −3.23488663651536556725432799365, −3.14978761307355954195363194354, −2.32061267747781760103864705700, −1.85042139776134295975923511880, −1.37234244543934425469486742792, −1.21806263970241569955795356761, −0.11977577245846564232444661477, 0.11977577245846564232444661477, 1.21806263970241569955795356761, 1.37234244543934425469486742792, 1.85042139776134295975923511880, 2.32061267747781760103864705700, 3.14978761307355954195363194354, 3.23488663651536556725432799365, 3.96009558417502354075370650205, 4.11360195887226238013533721709, 4.23142765820583406878319203939, 4.71314616758000778788612895226, 5.40454543706290499328685267004, 5.42685636239814820828882099831, 5.56497464752801068374341290045, 5.79331407752267733878280277254, 6.60916279389742401079830992922, 6.80222687969741269464423094826, 6.83875942143023981378150222726, 6.93545204156817504678880127090, 7.65691460233254945002174343382, 7.84204876759055956765639938549, 7.933881390920548303463260050649, 8.016155787730277047372686460105, 8.726663677637035073899286257173, 9.147633935025717020481531939898

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