Properties

Degree $4$
Conductor $254016$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 5·7-s − 6·11-s − 6·13-s + 4·17-s + 5·19-s − 4·23-s + 5·25-s + 8·29-s − 7·31-s + 10·35-s + 9·37-s + 4·41-s − 2·43-s + 2·47-s + 18·49-s + 8·53-s − 12·55-s − 10·61-s − 12·65-s + 15·67-s + 12·71-s + 11·73-s − 30·77-s − 79-s − 12·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.88·7-s − 1.80·11-s − 1.66·13-s + 0.970·17-s + 1.14·19-s − 0.834·23-s + 25-s + 1.48·29-s − 1.25·31-s + 1.69·35-s + 1.47·37-s + 0.624·41-s − 0.304·43-s + 0.291·47-s + 18/7·49-s + 1.09·53-s − 1.61·55-s − 1.28·61-s − 1.48·65-s + 1.83·67-s + 1.42·71-s + 1.28·73-s − 3.41·77-s − 0.112·79-s − 1.31·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(254016\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{504} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 254016,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35154\)
\(L(\frac12)\) \(\approx\) \(2.35154\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−10.94494209126543165424406328632, −10.87072075275514347279479372708, −10.10634625282412151359127238594, −9.852024814198108807289053012031, −9.645974251661760253846192525462, −8.890528885536642797470228626488, −8.178880345957336238189269793949, −8.098329698175852775873073983742, −7.50970555769538460703916113559, −7.34413989000089131159784952071, −6.61226052008171189603554239903, −5.70474726903762060798757274996, −5.30595750339914328143036925768, −5.27210085663573538402497997387, −4.66164764227298881821069566165, −4.06104830099305256895123266484, −2.87446724735790936959966011728, −2.55391115072584950497414244288, −1.94480432129522434333023636780, −0.972834040942710151842585743168, 0.972834040942710151842585743168, 1.94480432129522434333023636780, 2.55391115072584950497414244288, 2.87446724735790936959966011728, 4.06104830099305256895123266484, 4.66164764227298881821069566165, 5.27210085663573538402497997387, 5.30595750339914328143036925768, 5.70474726903762060798757274996, 6.61226052008171189603554239903, 7.34413989000089131159784952071, 7.50970555769538460703916113559, 8.098329698175852775873073983742, 8.178880345957336238189269793949, 8.890528885536642797470228626488, 9.645974251661760253846192525462, 9.852024814198108807289053012031, 10.10634625282412151359127238594, 10.87072075275514347279479372708, 10.94494209126543165424406328632

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