Properties

Label 4-3456-1.1-c1e2-0-6
Degree $4$
Conductor $3456$
Sign $1$
Analytic cond. $0.220357$
Root an. cond. $0.685144$
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-s + 9-s − 8·11-s − 4·13-s + 16·23-s − 6·25-s + 27-s − 8·33-s + 12·37-s − 4·39-s − 14·49-s − 8·59-s − 4·61-s + 16·69-s − 16·71-s + 20·73-s − 6·75-s + 81-s + 8·83-s + 4·97-s − 8·99-s + 24·107-s − 4·109-s + 12·111-s − 4·117-s + 26·121-s + 127-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 2.41·11-s − 1.10·13-s + 3.33·23-s − 6/5·25-s + 0.192·27-s − 1.39·33-s + 1.97·37-s − 0.640·39-s − 2·49-s − 1.04·59-s − 0.512·61-s + 1.92·69-s − 1.89·71-s + 2.34·73-s − 0.692·75-s + 1/9·81-s + 0.878·83-s + 0.406·97-s − 0.804·99-s + 2.32·107-s − 0.383·109-s + 1.13·111-s − 0.369·117-s + 2.36·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(0.220357\)
Root analytic conductor: \(0.685144\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3456} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8203437931\)
\(L(\frac12)\) \(\approx\) \(0.8203437931\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−12.79661702509897009228746961274, −12.32215863397544621963398389319, −11.17432181718975973992131540327, −11.02059068705223596291987853205, −10.16546244400809085298891139382, −9.656797755147918818405236990829, −9.065170023749915638598404079220, −8.173282349834764667604075089133, −7.62403207119791885504595168937, −7.26958585488936859387169650645, −6.14969218484336320739052245813, −5.01357758335939619070346679818, −4.83326855247570334999188527992, −3.14826068230388029805668446658, −2.51659494204614894722398853037, 2.51659494204614894722398853037, 3.14826068230388029805668446658, 4.83326855247570334999188527992, 5.01357758335939619070346679818, 6.14969218484336320739052245813, 7.26958585488936859387169650645, 7.62403207119791885504595168937, 8.173282349834764667604075089133, 9.065170023749915638598404079220, 9.656797755147918818405236990829, 10.16546244400809085298891139382, 11.02059068705223596291987853205, 11.17432181718975973992131540327, 12.32215863397544621963398389319, 12.79661702509897009228746961274

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