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Properties

Label 4-6e6-1.1-c1e2-0-7
Degree $4$
Conductor $46656$
Sign $1$
Analytic cond. $2.97482$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3·7-s + 5·11-s + 5·13-s + 4·17-s − 8·19-s − 23-s + 5·25-s − 9·29-s + 31-s − 3·35-s − 12·37-s + 3·41-s − 43-s − 3·47-s + 7·49-s − 4·53-s − 5·55-s + 11·59-s − 7·61-s − 5·65-s + 67-s − 8·71-s − 4·73-s + 15·77-s − 79-s + 83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s + 1.50·11-s + 1.38·13-s + 0.970·17-s − 1.83·19-s − 0.208·23-s + 25-s − 1.67·29-s + 0.179·31-s − 0.507·35-s − 1.97·37-s + 0.468·41-s − 0.152·43-s − 0.437·47-s + 49-s − 0.549·53-s − 0.674·55-s + 1.43·59-s − 0.896·61-s − 0.620·65-s + 0.122·67-s − 0.949·71-s − 0.468·73-s + 1.70·77-s − 0.112·79-s + 0.109·83-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 46656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.580576737\)
\(L(\frac12)\) \(\approx\) \(1.580576737\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_c
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.af_m
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) 2.23.b_aw
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_abe
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_abm
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.59.al_ck
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_aco
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_ada
83$C_2^2$ \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) 2.83.ab_ade
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.89.abk_ti
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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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

−12.33301225841225625353032989756, −12.08951976212825155123198909488, −11.54674992129962279971637330726, −11.19138873605400697127518679357, −10.54941577732228680779928163285, −10.52469225284218391935921330481, −9.482776687426013354812159507706, −8.999481552695027050128102820440, −8.428057998910143146019568767755, −8.421991505255167162476740288671, −7.47582765709396604650330576333, −7.12232559630798562910267941350, −6.21757808552297405855662949323, −6.08959875774201484024840710972, −5.10502125316642197607072094461, −4.55642019763593254059847043674, −3.67068864511486347733686858899, −3.62592093821060769785626162041, −2.03593968508812365837195667138, −1.30484125328406603231345639004, 1.30484125328406603231345639004, 2.03593968508812365837195667138, 3.62592093821060769785626162041, 3.67068864511486347733686858899, 4.55642019763593254059847043674, 5.10502125316642197607072094461, 6.08959875774201484024840710972, 6.21757808552297405855662949323, 7.12232559630798562910267941350, 7.47582765709396604650330576333, 8.421991505255167162476740288671, 8.428057998910143146019568767755, 8.999481552695027050128102820440, 9.482776687426013354812159507706, 10.52469225284218391935921330481, 10.54941577732228680779928163285, 11.19138873605400697127518679357, 11.54674992129962279971637330726, 12.08951976212825155123198909488, 12.33301225841225625353032989756

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