Properties

Label 4-2e6-1.1-c3e2-0-0
Degree $4$
Conductor $64$
Sign $1$
Analytic cond. $0.222798$
Root an. cond. $0.687033$
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  − 2·2-s − 4·4-s − 16·7-s + 24·8-s + 26·9-s + 32·14-s − 16·16-s − 28·17-s − 52·18-s − 304·23-s + 138·25-s + 64·28-s + 448·31-s − 160·32-s + 56·34-s − 104·36-s − 140·41-s + 608·46-s + 672·47-s − 494·49-s − 276·50-s − 384·56-s − 896·62-s − 416·63-s + 448·64-s + 112·68-s − 144·71-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.863·7-s + 1.06·8-s + 0.962·9-s + 0.610·14-s − 1/4·16-s − 0.399·17-s − 0.680·18-s − 2.75·23-s + 1.10·25-s + 0.431·28-s + 2.59·31-s − 0.883·32-s + 0.282·34-s − 0.481·36-s − 0.533·41-s + 1.94·46-s + 2.08·47-s − 1.44·49-s − 0.780·50-s − 0.916·56-s − 1.83·62-s − 0.831·63-s + 7/8·64-s + 0.199·68-s − 0.240·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Analytic conductor: \(0.222798\)
Root analytic conductor: \(0.687033\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 64,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4403280520\)
\(L(\frac12)\) \(\approx\) \(0.4403280520\)
\(L(\frac{5}{2})\) 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)$
bad2$C_2$ \( 1 + p T + p^{3} T^{2} \)
good3$C_2^2$ \( 1 - 26 T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 138 T^{2} + p^{6} T^{4} \)
7$C_2$ \( ( 1 + 8 T + p^{3} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 2410 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 1594 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 14 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 12346 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 152 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 23578 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 224 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 42058 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 70 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 33878 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 - 336 T + p^{3} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 296746 T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 125130 T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 444890 T^{2} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 571034 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 72 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 294 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 464 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 846522 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 266 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 994 T + p^{3} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.32328609141571665667068374535, −21.30376301435529161239728484600, −20.32217722439891987038930398912, −19.68782470900856199067021394123, −18.94139912399034712115163287588, −18.47465037908060946616208160159, −17.71952466704451273167168990255, −17.01365951247526908992298118503, −15.95486277328282696300256864193, −15.76870394334660738848907925024, −14.30533482400885281017161563303, −13.56118051764597305259579034934, −12.80093509323794710250940362899, −11.87562116271811096424835068025, −10.19856353228435028276780554201, −10.06406893318100641723738699310, −8.832485509020613984625872254045, −7.76460757569393509809552959748, −6.41965094825064143108164280420, −4.35379543989648781937842455545, 4.35379543989648781937842455545, 6.41965094825064143108164280420, 7.76460757569393509809552959748, 8.832485509020613984625872254045, 10.06406893318100641723738699310, 10.19856353228435028276780554201, 11.87562116271811096424835068025, 12.80093509323794710250940362899, 13.56118051764597305259579034934, 14.30533482400885281017161563303, 15.76870394334660738848907925024, 15.95486277328282696300256864193, 17.01365951247526908992298118503, 17.71952466704451273167168990255, 18.47465037908060946616208160159, 18.94139912399034712115163287588, 19.68782470900856199067021394123, 20.32217722439891987038930398912, 21.30376301435529161239728484600, 22.32328609141571665667068374535

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