Properties

Label 4-12e4-1.1-c1e2-0-2
Degree $4$
Conductor $20736$
Sign $1$
Analytic cond. $1.32214$
Root an. cond. $1.07230$
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 − 3·9-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 − 3·45-s − 3·47-s + 7·49-s + 4·53-s + 5·55-s + 11·59-s − 7·61-s + 9·63-s + 5·65-s − 67-s − 8·71-s − 4·73-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s − 9-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 − 0.447·45-s − 0.437·47-s + 49-s + 0.549·53-s + 0.674·55-s + 1.43·59-s − 0.896·61-s + 1.13·63-s + 0.620·65-s − 0.122·67-s − 0.949·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20736\)    =    \(2^{8} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.32214\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{144} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20736,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.185432553\)
\(L(\frac12)\) \(\approx\) \(1.185432553\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−13.69769288080972841376805078173, −12.91711883807172063042696979752, −12.17827624041516065345586035141, −11.94210221340455455198242576996, −11.37394362891619155388960776176, −10.89080209585435461437045072888, −10.20837879914991791089988814963, −9.780415696448317845299387661805, −9.110252047570485475361892622153, −8.610030166282449061877628348495, −8.608424029237451573602788670117, −7.33291629688830953280640226187, −6.64745874278211041824038420645, −6.49326918264950109464074853900, −5.76344677891560334785169615219, −5.17698809392735266441884198534, −4.10744132624351610669430925227, −3.39625860626815687212383917190, −2.83772857287000543069694881152, −1.31259399580731222414321464551, 1.31259399580731222414321464551, 2.83772857287000543069694881152, 3.39625860626815687212383917190, 4.10744132624351610669430925227, 5.17698809392735266441884198534, 5.76344677891560334785169615219, 6.49326918264950109464074853900, 6.64745874278211041824038420645, 7.33291629688830953280640226187, 8.608424029237451573602788670117, 8.610030166282449061877628348495, 9.110252047570485475361892622153, 9.780415696448317845299387661805, 10.20837879914991791089988814963, 10.89080209585435461437045072888, 11.37394362891619155388960776176, 11.94210221340455455198242576996, 12.17827624041516065345586035141, 12.91711883807172063042696979752, 13.69769288080972841376805078173

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