Properties

Label 4-882e2-1.1-c3e2-0-0
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 − 14·5-s + 8·8-s + 28·10-s − 28·11-s + 36·13-s − 16·16-s + 74·17-s − 80·19-s + 56·22-s − 112·23-s + 125·25-s − 72·26-s − 380·29-s − 72·31-s − 148·34-s + 346·37-s + 160·38-s − 112·40-s − 324·41-s − 824·43-s + 224·46-s + 24·47-s − 250·50-s + 318·53-s + 392·55-s + 760·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.25·5-s + 0.353·8-s + 0.885·10-s − 0.767·11-s + 0.768·13-s − 1/4·16-s + 1.05·17-s − 0.965·19-s + 0.542·22-s − 1.01·23-s + 25-s − 0.543·26-s − 2.43·29-s − 0.417·31-s − 0.746·34-s + 1.53·37-s + 0.683·38-s − 0.442·40-s − 1.23·41-s − 2.92·43-s + 0.717·46-s + 0.0744·47-s − 0.707·50-s + 0.824·53-s + 0.961·55-s + 1.72·58-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

\(L(2)\) \(\approx\) \(0.04855143244\)
\(L(\frac12)\) \(\approx\) \(0.04855143244\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 14 T + 71 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 28 T - 547 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 18 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 74 T + 563 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 80 T - 459 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 112 T + 377 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 190 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 72 T - 24607 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 346 T + 69063 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 162 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 412 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 24 T - 103247 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 6 p T - 17 p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 200 T - 165379 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 198 T - 187777 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 716 T + 211893 T^{2} - 716 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 392 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 538 T - 99573 T^{2} + 538 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 240 T - 435439 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 1072 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 810 T - 48869 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 1354 T + p^{3} T^{2} )^{2} \)
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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

−9.861393497598786178970393283272, −9.606478394605896778823228647951, −9.071233578554283400808104945153, −8.487162287820145737908884077916, −8.349015558766799485354970800141, −7.84602910106019647461726148471, −7.66167613167549087394831792121, −7.18290496778931765504583433834, −6.61957134208547286845542164106, −6.12487352319705988304288386826, −5.58437660317652174596301117371, −5.12879999039638708641614389843, −4.57818685151988653673035464103, −3.98858041896671734672985763494, −3.49665243315996359222679105096, −3.32963772638645967457239571306, −2.23352628283354131143899657181, −1.78455760980068551552377119461, −0.953689675696556563660130204592, −0.07849625838598364591452362339, 0.07849625838598364591452362339, 0.953689675696556563660130204592, 1.78455760980068551552377119461, 2.23352628283354131143899657181, 3.32963772638645967457239571306, 3.49665243315996359222679105096, 3.98858041896671734672985763494, 4.57818685151988653673035464103, 5.12879999039638708641614389843, 5.58437660317652174596301117371, 6.12487352319705988304288386826, 6.61957134208547286845542164106, 7.18290496778931765504583433834, 7.66167613167549087394831792121, 7.84602910106019647461726148471, 8.349015558766799485354970800141, 8.487162287820145737908884077916, 9.071233578554283400808104945153, 9.606478394605896778823228647951, 9.861393497598786178970393283272

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