Properties

Label 4-980e2-1.1-c1e2-0-13
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 3·9-s − 3·11-s + 2·13-s + 15-s − 3·17-s + 2·19-s + 6·23-s + 8·27-s − 18·29-s + 8·31-s − 3·33-s + 10·37-s + 2·39-s + 4·43-s + 3·45-s − 3·47-s − 3·51-s − 3·55-s + 2·57-s + 12·59-s + 8·61-s + 2·65-s − 8·67-s + 6·69-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 9-s − 0.904·11-s + 0.554·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s + 1.25·23-s + 1.53·27-s − 3.34·29-s + 1.43·31-s − 0.522·33-s + 1.64·37-s + 0.320·39-s + 0.609·43-s + 0.447·45-s − 0.437·47-s − 0.420·51-s − 0.404·55-s + 0.264·57-s + 1.56·59-s + 1.02·61-s + 0.248·65-s − 0.977·67-s + 0.722·69-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.047381068\)
\(L(\frac12)\) \(\approx\) \(3.047381068\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
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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

−10.03077824192863008076771116252, −9.743090891685292307546137647562, −9.410702100232467724326086596023, −9.062667533515878684049021687528, −8.517263805985110385723093610809, −8.194868196530131526967800745369, −7.62767629169851764066551470561, −7.38095680544537130193362580007, −6.86955817536010857967183384958, −6.46679365889822719237625963845, −5.88003643847922287329289675124, −5.46890453147666287549010541676, −4.88775740636797138867394016278, −4.57426435890614877901453642291, −3.76178797567082590955893838189, −3.58188869311409331164942614819, −2.55001106960615566087246522037, −2.49136670286715251620276324972, −1.59004638298352418964094952922, −0.822239012411896174393442058562, 0.822239012411896174393442058562, 1.59004638298352418964094952922, 2.49136670286715251620276324972, 2.55001106960615566087246522037, 3.58188869311409331164942614819, 3.76178797567082590955893838189, 4.57426435890614877901453642291, 4.88775740636797138867394016278, 5.46890453147666287549010541676, 5.88003643847922287329289675124, 6.46679365889822719237625963845, 6.86955817536010857967183384958, 7.38095680544537130193362580007, 7.62767629169851764066551470561, 8.194868196530131526967800745369, 8.517263805985110385723093610809, 9.062667533515878684049021687528, 9.410702100232467724326086596023, 9.743090891685292307546137647562, 10.03077824192863008076771116252

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