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Properties

Label 4-12e6-1.1-c1e2-0-10
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $190.388$
Root an. cond. $3.71458$
Motivic weight $1$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(190.388\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2985984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.373783203\)
\(L(\frac12)\) \(\approx\) \(2.373783203\)
\(L(\frac{3}{2})\) 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.ab_ae
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_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.ab_aw
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_abe
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_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.ad_abm
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_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.ah_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.ai_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.ab_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

−9.397500391788558059738583764568, −9.230954184198371637554746575224, −8.871480675738745095058805941503, −8.444607004065448806889327827513, −7.82638695898537473534990005331, −7.65008371866623034981955284199, −6.86890953461105974350371490219, −6.56682407550779397664901035426, −6.55321300412670306495841411810, −6.07485660590617473361141374650, −5.39614871774321926353077742452, −5.12762141093597702548674493333, −4.42049811445718753972479674886, −4.13375891859639884914254861608, −3.68690669879141820979525804558, −3.01747653740029143276025991226, −2.47296833887514109004886028854, −2.27001762369925471968414242323, −1.17967172393650958467300979696, −0.65793870774978760978756506439, 0.65793870774978760978756506439, 1.17967172393650958467300979696, 2.27001762369925471968414242323, 2.47296833887514109004886028854, 3.01747653740029143276025991226, 3.68690669879141820979525804558, 4.13375891859639884914254861608, 4.42049811445718753972479674886, 5.12762141093597702548674493333, 5.39614871774321926353077742452, 6.07485660590617473361141374650, 6.55321300412670306495841411810, 6.56682407550779397664901035426, 6.86890953461105974350371490219, 7.65008371866623034981955284199, 7.82638695898537473534990005331, 8.444607004065448806889327827513, 8.871480675738745095058805941503, 9.230954184198371637554746575224, 9.397500391788558059738583764568

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