Properties

Label 4-456e2-1.1-c1e2-0-22
Degree $4$
Conductor $207936$
Sign $1$
Analytic cond. $13.2581$
Root an. cond. $1.90818$
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  + 2·3-s + 5-s + 7-s + 3·9-s + 7·11-s − 2·13-s + 2·15-s + 17-s + 2·19-s + 2·21-s + 10·23-s − 5·25-s + 4·27-s − 4·31-s + 14·33-s + 35-s − 16·37-s − 4·39-s − 2·41-s + 43-s + 3·45-s − 47-s − 9·49-s + 2·51-s − 8·53-s + 7·55-s + 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.377·7-s + 9-s + 2.11·11-s − 0.554·13-s + 0.516·15-s + 0.242·17-s + 0.458·19-s + 0.436·21-s + 2.08·23-s − 25-s + 0.769·27-s − 0.718·31-s + 2.43·33-s + 0.169·35-s − 2.63·37-s − 0.640·39-s − 0.312·41-s + 0.152·43-s + 0.447·45-s − 0.145·47-s − 9/7·49-s + 0.280·51-s − 1.09·53-s + 0.943·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(207936\)    =    \(2^{6} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(13.2581\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{456} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 207936,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.440733748\)
\(L(\frac12)\) \(\approx\) \(3.440733748\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T + 82 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 56 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 5 T - 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−11.24540968226618014154682550052, −10.94686860783810601458415392316, −10.04551982372473329199643082346, −9.985716414794461087196564862141, −9.270778395815247124693524571105, −9.205059440403557799239543917143, −8.511267098641249115618900934208, −8.470762069674737448807889307182, −7.40195414680424503888986442287, −7.36924501536222123211794169192, −6.62895753852762574295251684134, −6.49618163820918364685782914467, −5.35681992211034713205325762601, −5.21840378450538605319539538173, −4.40683430383325362586312797855, −3.71097266423840225411789316701, −3.45450757029290982715145021911, −2.65770680178761096011354409403, −1.74194842960803647557036580265, −1.35376246294913576804396460802, 1.35376246294913576804396460802, 1.74194842960803647557036580265, 2.65770680178761096011354409403, 3.45450757029290982715145021911, 3.71097266423840225411789316701, 4.40683430383325362586312797855, 5.21840378450538605319539538173, 5.35681992211034713205325762601, 6.49618163820918364685782914467, 6.62895753852762574295251684134, 7.36924501536222123211794169192, 7.40195414680424503888986442287, 8.470762069674737448807889307182, 8.511267098641249115618900934208, 9.205059440403557799239543917143, 9.270778395815247124693524571105, 9.985716414794461087196564862141, 10.04551982372473329199643082346, 10.94686860783810601458415392316, 11.24540968226618014154682550052

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