Properties

Label 4-882e2-1.1-c3e2-0-1
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

\(L(2)\) \(\approx\) \(0.2146348919\)
\(L(\frac12)\) \(\approx\) \(0.2146348919\)
\(L(\frac{5}{2})\) 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

−10.00170122087802948614579645946, −9.774911121411751586907822049782, −9.300491288941149491459526589129, −8.642471081015980497833945130408, −8.477209533358224688985791900048, −7.68455546248623220390012617973, −7.68447865745675963982812829219, −6.89365759487603518281850144934, −6.73736918946184096340636719163, −5.93721407569318898016079678301, −5.60100855273159081277578264392, −5.31457054639016316857282121875, −4.68055355915688163262178823103, −4.18864977934043127024094913692, −3.55752672101578901233160434951, −2.60850489057279171447232983818, −2.49323570940076165265261769654, −1.73621844186879382633949984982, −1.25762082007087556917051727696, −0.13465366230201023326126197245, 0.13465366230201023326126197245, 1.25762082007087556917051727696, 1.73621844186879382633949984982, 2.49323570940076165265261769654, 2.60850489057279171447232983818, 3.55752672101578901233160434951, 4.18864977934043127024094913692, 4.68055355915688163262178823103, 5.31457054639016316857282121875, 5.60100855273159081277578264392, 5.93721407569318898016079678301, 6.73736918946184096340636719163, 6.89365759487603518281850144934, 7.68447865745675963982812829219, 7.68455546248623220390012617973, 8.477209533358224688985791900048, 8.642471081015980497833945130408, 9.300491288941149491459526589129, 9.774911121411751586907822049782, 10.00170122087802948614579645946

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