Properties

Label 4-6080e2-1.1-c1e2-0-3
Degree $4$
Conductor $36966400$
Sign $1$
Analytic cond. $2357.00$
Root an. cond. $6.96771$
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 − 2·5-s + 7-s − 9-s + 8·11-s + 13-s + 2·15-s + 11·17-s − 2·19-s − 21-s − 3·23-s + 3·25-s − 29-s + 2·31-s − 8·33-s − 2·35-s + 12·37-s − 39-s + 8·41-s − 14·43-s + 2·45-s + 4·47-s − 9·49-s − 11·51-s + 5·53-s − 16·55-s + 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s − 1/3·9-s + 2.41·11-s + 0.277·13-s + 0.516·15-s + 2.66·17-s − 0.458·19-s − 0.218·21-s − 0.625·23-s + 3/5·25-s − 0.185·29-s + 0.359·31-s − 1.39·33-s − 0.338·35-s + 1.97·37-s − 0.160·39-s + 1.24·41-s − 2.13·43-s + 0.298·45-s + 0.583·47-s − 9/7·49-s − 1.54·51-s + 0.686·53-s − 2.15·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36966400\)    =    \(2^{12} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2357.00\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36966400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.027665421\)
\(L(\frac12)\) \(\approx\) \(3.027665421\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 130 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 9 T + 128 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 198 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−8.141745790441977211557128741325, −7.79732171892929551856498387775, −7.52542176653988580776767698400, −7.44738393459509477146804517097, −6.65297236531291480398749634789, −6.38966054047371223141324720467, −6.08457500745039539164562322487, −6.03220379626063860303367780049, −5.30432974143354259300411314089, −5.08009440961062357203542543591, −4.59446104319726375228505470424, −4.03977140890771832602713617996, −4.00844173957005114716363647797, −3.46860416874096897819261564049, −3.20008156914533495589635779886, −2.69896894035271525170081604366, −1.80315910860449273772587589851, −1.54229286140479484424134480798, −0.897917088678288154552105722534, −0.61643329933232040631517269475, 0.61643329933232040631517269475, 0.897917088678288154552105722534, 1.54229286140479484424134480798, 1.80315910860449273772587589851, 2.69896894035271525170081604366, 3.20008156914533495589635779886, 3.46860416874096897819261564049, 4.00844173957005114716363647797, 4.03977140890771832602713617996, 4.59446104319726375228505470424, 5.08009440961062357203542543591, 5.30432974143354259300411314089, 6.03220379626063860303367780049, 6.08457500745039539164562322487, 6.38966054047371223141324720467, 6.65297236531291480398749634789, 7.44738393459509477146804517097, 7.52542176653988580776767698400, 7.79732171892929551856498387775, 8.141745790441977211557128741325

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