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Properties

Label 4-6e6-1.1-c1e2-0-3
Degree $4$
Conductor $46656$
Sign $1$
Analytic cond. $2.97482$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 5·7-s − 10·14-s − 4·16-s − 4·17-s + 8·23-s − 3·25-s + 10·28-s + 4·31-s + 8·32-s + 8·34-s + 4·41-s − 16·46-s + 12·47-s + 7·49-s + 6·50-s − 8·62-s − 8·64-s − 8·68-s − 12·71-s + 15·73-s + 2·79-s − 8·82-s − 20·89-s + 16·92-s − 24·94-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.88·7-s − 2.67·14-s − 16-s − 0.970·17-s + 1.66·23-s − 3/5·25-s + 1.88·28-s + 0.718·31-s + 1.41·32-s + 1.37·34-s + 0.624·41-s − 2.35·46-s + 1.75·47-s + 49-s + 0.848·50-s − 1.01·62-s − 64-s − 0.970·68-s − 1.42·71-s + 1.75·73-s + 0.225·79-s − 0.883·82-s − 2.11·89-s + 1.66·92-s − 2.47·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46656\)    =    \(2^{6} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(2.97482\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 46656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8195001340\)
\(L(\frac12)\) \(\approx\) \(0.8195001340\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.7.af_s
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.a_r
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.e_c
19$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.19.a_f
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.31.ae_cf
37$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.37.a_r
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.ae_by
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.43.a_bm
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.am_dq
53$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.53.a_acj
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.59.a_g
61$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.61.a_ax
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.a_abj
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.m_fm
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.73.ap_hs
79$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.ac_p
83$C_2^2$ \( 1 - 99 T^{2} + p^{2} T^{4} \) 2.83.a_adv
89$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.u_ji
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.97.n_ii
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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

−10.13406517822517540415991980029, −9.508945804322141365448729514116, −9.043275478727530859761826686634, −8.581411912559862447400176029117, −8.246244534258173635557406789226, −7.68677947931311725879504343322, −7.24528488548728832255956470863, −6.75125048746113267444674472732, −5.91420727417963782021867240501, −5.11913315620451330754535000136, −4.63596909164702535252832687134, −4.09364766916069031941739700137, −2.73020780463111989951793154185, −1.94893279693844340479685212296, −1.08464236819152012197175855629, 1.08464236819152012197175855629, 1.94893279693844340479685212296, 2.73020780463111989951793154185, 4.09364766916069031941739700137, 4.63596909164702535252832687134, 5.11913315620451330754535000136, 5.91420727417963782021867240501, 6.75125048746113267444674472732, 7.24528488548728832255956470863, 7.68677947931311725879504343322, 8.246244534258173635557406789226, 8.581411912559862447400176029117, 9.043275478727530859761826686634, 9.508945804322141365448729514116, 10.13406517822517540415991980029

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