Properties

Label 4-574e2-1.1-c1e2-0-11
Degree $4$
Conductor $329476$
Sign $1$
Analytic cond. $21.0076$
Root an. cond. $2.14089$
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  + 2·2-s + 4·3-s + 3·4-s + 2·5-s + 8·6-s + 2·7-s + 4·8-s + 6·9-s + 4·10-s − 2·11-s + 12·12-s − 4·13-s + 4·14-s + 8·15-s + 5·16-s + 12·18-s + 6·20-s + 8·21-s − 4·22-s − 4·23-s + 16·24-s − 4·25-s − 8·26-s − 4·27-s + 6·28-s + 2·29-s + 16·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 3/2·4-s + 0.894·5-s + 3.26·6-s + 0.755·7-s + 1.41·8-s + 2·9-s + 1.26·10-s − 0.603·11-s + 3.46·12-s − 1.10·13-s + 1.06·14-s + 2.06·15-s + 5/4·16-s + 2.82·18-s + 1.34·20-s + 1.74·21-s − 0.852·22-s − 0.834·23-s + 3.26·24-s − 4/5·25-s − 1.56·26-s − 0.769·27-s + 1.13·28-s + 0.371·29-s + 2.92·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(329476\)    =    \(2^{2} \cdot 7^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(21.0076\)
Root analytic conductor: \(2.14089\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 329476,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.70258435\)
\(L(\frac12)\) \(\approx\) \(10.70258435\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
41$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 2 T - 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T - 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.83414966746262515332150361368, −10.57153349817273593405619244112, −9.813132539758024597169888458094, −9.794406286648013319719209322747, −9.066189793075234107295096885690, −8.798880276564951093933163698816, −8.065298708873849097525483999286, −7.967017025517771098830697611375, −7.29028196753718996087160733850, −7.26569317579054567255770987474, −6.12366354484206471875733356811, −5.93172699425919030021950594286, −5.19300327966827695169554689757, −4.97304994045795065550684133973, −4.01036689106736926473861118481, −3.83849670089299173961744248044, −3.08476137398688017237958118800, −2.47520088288264753241437176258, −2.23595805020013558835760984666, −1.79538789664999522192870703826, 1.79538789664999522192870703826, 2.23595805020013558835760984666, 2.47520088288264753241437176258, 3.08476137398688017237958118800, 3.83849670089299173961744248044, 4.01036689106736926473861118481, 4.97304994045795065550684133973, 5.19300327966827695169554689757, 5.93172699425919030021950594286, 6.12366354484206471875733356811, 7.26569317579054567255770987474, 7.29028196753718996087160733850, 7.967017025517771098830697611375, 8.065298708873849097525483999286, 8.798880276564951093933163698816, 9.066189793075234107295096885690, 9.794406286648013319719209322747, 9.813132539758024597169888458094, 10.57153349817273593405619244112, 10.83414966746262515332150361368

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