Properties

Label 4-12e4-1.1-c1e2-0-17
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  + 3·3-s + 2·7-s + 6·9-s − 3·11-s − 2·13-s − 6·17-s + 2·19-s + 6·21-s − 6·23-s + 5·25-s + 9·27-s − 6·29-s − 4·31-s − 9·33-s − 8·37-s − 6·39-s − 9·41-s − 43-s − 6·47-s + 7·49-s − 18·51-s + 24·53-s + 6·57-s + 3·59-s − 8·61-s + 12·63-s + 5·67-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.755·7-s + 2·9-s − 0.904·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.30·21-s − 1.25·23-s + 25-s + 1.73·27-s − 1.11·29-s − 0.718·31-s − 1.56·33-s − 1.31·37-s − 0.960·39-s − 1.40·41-s − 0.152·43-s − 0.875·47-s + 49-s − 2.52·51-s + 3.29·53-s + 0.794·57-s + 0.390·59-s − 1.02·61-s + 1.51·63-s + 0.610·67-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

\(L(1)\) \(\approx\) \(1.960948954\)
\(L(\frac12)\) \(\approx\) \(1.960948954\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−13.29412028177970020405968736759, −13.26753273033277262417649451966, −12.32959609175909797451294911946, −12.08741449551169059831330125042, −11.21134912092071582860274237943, −10.74238199510750267337032779634, −10.20755662618736127992191657134, −9.699742114448970930952488412869, −8.985259579193792400133529119334, −8.773992186911460946511155514314, −7.957456630593232112232971517021, −7.914504181237864992656505890458, −7.00489071735715296868076371624, −6.72808141967301058242909305741, −5.31075452569347015793858506264, −5.05711936670991334027722453981, −4.00947033884517203424427915554, −3.53519034210908129381929486333, −2.35108135638940597967821092355, −2.04901078629379074055897362975, 2.04901078629379074055897362975, 2.35108135638940597967821092355, 3.53519034210908129381929486333, 4.00947033884517203424427915554, 5.05711936670991334027722453981, 5.31075452569347015793858506264, 6.72808141967301058242909305741, 7.00489071735715296868076371624, 7.914504181237864992656505890458, 7.957456630593232112232971517021, 8.773992186911460946511155514314, 8.985259579193792400133529119334, 9.699742114448970930952488412869, 10.20755662618736127992191657134, 10.74238199510750267337032779634, 11.21134912092071582860274237943, 12.08741449551169059831330125042, 12.32959609175909797451294911946, 13.26753273033277262417649451966, 13.29412028177970020405968736759

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