Properties

Label 4-882e2-1.1-c3e2-0-25
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $2708.12$
Root an. cond. $7.21385$
Motivic weight $3$
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 + 6·5-s + 8·8-s − 12·10-s − 30·11-s − 4·13-s − 16·16-s + 66·17-s − 52·19-s + 60·22-s − 114·23-s + 125·25-s + 8·26-s + 144·29-s − 196·31-s − 132·34-s + 286·37-s + 104·38-s + 48·40-s + 756·41-s + 328·43-s + 228·46-s − 228·47-s − 250·50-s + 348·53-s − 180·55-s − 288·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.536·5-s + 0.353·8-s − 0.379·10-s − 0.822·11-s − 0.0853·13-s − 1/4·16-s + 0.941·17-s − 0.627·19-s + 0.581·22-s − 1.03·23-s + 25-s + 0.0603·26-s + 0.922·29-s − 1.13·31-s − 0.665·34-s + 1.27·37-s + 0.443·38-s + 0.189·40-s + 2.87·41-s + 1.16·43-s + 0.730·46-s − 0.707·47-s − 0.707·50-s + 0.901·53-s − 0.441·55-s − 0.652·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2708.12\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.469460038\)
\(L(\frac12)\) \(\approx\) \(2.469460038\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 66 T - 557 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 52 T - 4155 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 114 T + 829 T^{2} + 114 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 72 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 196 T + 8625 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 286 T + 31143 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 378 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 228 T - 51839 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 348 T - 27773 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 348 T - 84275 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 106 T - 215745 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 630 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1042 T + 696747 T^{2} + 1042 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 88 T - 485295 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 1440 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1374 T + 1182907 T^{2} - 1374 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 34 T + p^{3} T^{2} )^{2} \)
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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

−9.911747274767855155764208235351, −9.538538967930463187312886892441, −9.020389069453734014911148294196, −8.977295028690960647299524556894, −8.127124506809661961081667218024, −7.960002754200240815875550248986, −7.51447013246395776466650983737, −7.21936722070184793903233450246, −6.42536721549335911626879342454, −6.00061347457366707799859806153, −5.80989874921359243988112610296, −5.13686654841852371471753554061, −4.58023555346908045159527297408, −4.27023901324165010245570402258, −3.47223936446646124699993772583, −2.92515121386079999367154381441, −2.24259836233202707629147241309, −1.92950797605858937262719438194, −0.76325493790958924150143379966, −0.69072460595655420548338110325, 0.69072460595655420548338110325, 0.76325493790958924150143379966, 1.92950797605858937262719438194, 2.24259836233202707629147241309, 2.92515121386079999367154381441, 3.47223936446646124699993772583, 4.27023901324165010245570402258, 4.58023555346908045159527297408, 5.13686654841852371471753554061, 5.80989874921359243988112610296, 6.00061347457366707799859806153, 6.42536721549335911626879342454, 7.21936722070184793903233450246, 7.51447013246395776466650983737, 7.960002754200240815875550248986, 8.127124506809661961081667218024, 8.977295028690960647299524556894, 9.020389069453734014911148294196, 9.538538967930463187312886892441, 9.911747274767855155764208235351

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