Properties

Label 4-39e4-1.1-c1e2-0-2
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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  + 3·4-s + 5·16-s − 14·17-s − 12·23-s + 9·25-s + 2·29-s − 12·43-s + 10·49-s + 18·53-s + 2·61-s + 3·64-s − 42·68-s − 8·79-s − 36·92-s + 27·100-s + 6·101-s − 12·103-s + 12·107-s + 30·113-s + 6·116-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 3/2·4-s + 5/4·16-s − 3.39·17-s − 2.50·23-s + 9/5·25-s + 0.371·29-s − 1.82·43-s + 10/7·49-s + 2.47·53-s + 0.256·61-s + 3/8·64-s − 5.09·68-s − 0.900·79-s − 3.75·92-s + 2.69·100-s + 0.597·101-s − 1.18·103-s + 1.16·107-s + 2.82·113-s + 0.557·116-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.347098059\)
\(L(\frac12)\) \(\approx\) \(2.347098059\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.721887559180964113611639948170, −9.228776181286645982006838089994, −8.625023972237513957465045706406, −8.494602370793332922917974930310, −8.312067228181774045852480533780, −7.31369343542435591955382439571, −7.25333726366425297980684803423, −6.85388800529815958650975446941, −6.49942745626158974401964656137, −6.07086546878824947133683665517, −5.85748183945164244449716008706, −5.00366168775714292143470087061, −4.70053793297145509188274322382, −4.05968353687517304113582246009, −3.82868852592147685335232962155, −2.89081336322744422982611162073, −2.56021628755197362769519055496, −2.01715652076063376273312109464, −1.79178813249096849552527210282, −0.55111203215314787658022323502, 0.55111203215314787658022323502, 1.79178813249096849552527210282, 2.01715652076063376273312109464, 2.56021628755197362769519055496, 2.89081336322744422982611162073, 3.82868852592147685335232962155, 4.05968353687517304113582246009, 4.70053793297145509188274322382, 5.00366168775714292143470087061, 5.85748183945164244449716008706, 6.07086546878824947133683665517, 6.49942745626158974401964656137, 6.85388800529815958650975446941, 7.25333726366425297980684803423, 7.31369343542435591955382439571, 8.312067228181774045852480533780, 8.494602370793332922917974930310, 8.625023972237513957465045706406, 9.228776181286645982006838089994, 9.721887559180964113611639948170

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