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Properties

Label 4-12e6-1.1-c1e2-0-15
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

\(L(1)\) \(\approx\) \(3.161153475\)
\(L(\frac12)\) \(\approx\) \(3.161153475\)
\(L(\frac{3}{2})\) 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.ad_c
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_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.ai_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.aj_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.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.ab_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.ae_eg
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.59.l_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.b_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.b_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

−9.543731381351568004749203531654, −9.125793295497294670814919804552, −8.814670618367569646028829830722, −7.991496866410054745167653272593, −7.914012287137553660015379919875, −7.79915518866265837920495041644, −7.11280994275382709040086144947, −7.02962111817897195154316627126, −6.00948852343102586575430579563, −5.95753065724749892998102676273, −5.35236108842587891873169991266, −4.95739148008411474292527925225, −4.69724642288974768102775052680, −4.45196926425254274734800207554, −3.31861385924957063349058800128, −3.11963436013871109334904510300, −2.42785416843290223160096242374, −2.24771920356935473319912296767, −1.19909660639261215221639309868, −0.76882466295312432264096735695, 0.76882466295312432264096735695, 1.19909660639261215221639309868, 2.24771920356935473319912296767, 2.42785416843290223160096242374, 3.11963436013871109334904510300, 3.31861385924957063349058800128, 4.45196926425254274734800207554, 4.69724642288974768102775052680, 4.95739148008411474292527925225, 5.35236108842587891873169991266, 5.95753065724749892998102676273, 6.00948852343102586575430579563, 7.02962111817897195154316627126, 7.11280994275382709040086144947, 7.79915518866265837920495041644, 7.914012287137553660015379919875, 7.991496866410054745167653272593, 8.814670618367569646028829830722, 9.125793295497294670814919804552, 9.543731381351568004749203531654

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