Properties

Label 4-9280e2-1.1-c1e2-0-1
Degree $4$
Conductor $86118400$
Sign $1$
Analytic cond. $5490.98$
Root an. cond. $8.60820$
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  − 4·3-s − 2·5-s + 4·7-s + 6·9-s − 4·11-s + 4·13-s + 8·15-s − 4·19-s − 16·21-s + 12·23-s + 3·25-s + 4·27-s − 2·29-s + 4·31-s + 16·33-s − 8·35-s − 16·39-s − 12·41-s − 12·43-s − 12·45-s + 12·47-s + 6·49-s − 4·53-s + 8·55-s + 16·57-s − 4·61-s + 24·63-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s + 1.51·7-s + 2·9-s − 1.20·11-s + 1.10·13-s + 2.06·15-s − 0.917·19-s − 3.49·21-s + 2.50·23-s + 3/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s + 2.78·33-s − 1.35·35-s − 2.56·39-s − 1.87·41-s − 1.82·43-s − 1.78·45-s + 1.75·47-s + 6/7·49-s − 0.549·53-s + 1.07·55-s + 2.11·57-s − 0.512·61-s + 3.02·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86118400\)    =    \(2^{12} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5490.98\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86118400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7733907368\)
\(L(\frac12)\) \(\approx\) \(0.7733907368\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.74393887510783984474495163827, −7.59932487812108149441765818527, −6.96807598364557694145082935062, −6.87783239425211696312712032527, −6.37127013998031901860124061861, −6.32090530167298415650612728172, −5.63453358879925384067071873644, −5.33560236191220038186171165641, −5.25434435111077369660705481209, −4.85777586287927416587578480845, −4.65462151378253281909667432828, −4.30016150101849990150943356883, −3.76617100089857571109271279575, −3.24646832875909124368767226650, −2.88042051635152943426666082430, −2.45067026380256152716281166188, −1.48863754557940030231728729723, −1.48219368717141098336829303026, −0.70232906168630852487091684958, −0.36353875978956403783882246030, 0.36353875978956403783882246030, 0.70232906168630852487091684958, 1.48219368717141098336829303026, 1.48863754557940030231728729723, 2.45067026380256152716281166188, 2.88042051635152943426666082430, 3.24646832875909124368767226650, 3.76617100089857571109271279575, 4.30016150101849990150943356883, 4.65462151378253281909667432828, 4.85777586287927416587578480845, 5.25434435111077369660705481209, 5.33560236191220038186171165641, 5.63453358879925384067071873644, 6.32090530167298415650612728172, 6.37127013998031901860124061861, 6.87783239425211696312712032527, 6.96807598364557694145082935062, 7.59932487812108149441765818527, 7.74393887510783984474495163827

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