Properties

Label 4-882e2-1.1-c2e2-0-0
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $577.573$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2·13-s + 4·16-s − 46·19-s + 32·25-s − 94·31-s − 110·37-s + 46·43-s − 4·52-s − 208·61-s − 8·64-s − 194·67-s − 130·73-s + 92·76-s + 226·79-s − 208·97-s − 64·100-s − 238·103-s − 98·109-s + 80·121-s + 188·124-s + 127-s + 131-s + 137-s + 139-s + 220·148-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/13·13-s + 1/4·16-s − 2.42·19-s + 1.27·25-s − 3.03·31-s − 2.97·37-s + 1.06·43-s − 0.0769·52-s − 3.40·61-s − 1/8·64-s − 2.89·67-s − 1.78·73-s + 1.21·76-s + 2.86·79-s − 2.14·97-s − 0.639·100-s − 2.31·103-s − 0.899·109-s + 0.661·121-s + 1.51·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.48·148-s + 0.00671·149-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(577.573\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.07433264945\)
\(L(\frac12)\) \(\approx\) \(0.07433264945\)
\(L(2)\) 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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 32 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 80 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 23 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 770 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 530 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 47 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 55 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 1184 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4400 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 3026 T^{2} + p^{4} T^{4} \)
59$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 + 104 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 97 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 560 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 65 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 113 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 12896 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 + 2590 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 104 T + p^{2} T^{2} )^{2} \)
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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

−10.55453481640419041740529738903, −9.505713386376568673210476808105, −9.222054175908024045832977661098, −8.910182480217836198499580298705, −8.607746552631514427123925111892, −8.176084068298713639108636445033, −7.57262719675420339795077380394, −7.04640739208492592152124389790, −6.92654382232964623444551127333, −6.10542686613735438204102033904, −5.89210002556530629649368849919, −5.29473893408680961965871669588, −4.76948962132770955315423845377, −4.35557174064098322875808167754, −3.83706515057428053077967521764, −3.32598395922472054956012873035, −2.72177873511208712586506191645, −1.80454228260531207055942104922, −1.54796115841814924399104948350, −0.087164595751584953522891153827, 0.087164595751584953522891153827, 1.54796115841814924399104948350, 1.80454228260531207055942104922, 2.72177873511208712586506191645, 3.32598395922472054956012873035, 3.83706515057428053077967521764, 4.35557174064098322875808167754, 4.76948962132770955315423845377, 5.29473893408680961965871669588, 5.89210002556530629649368849919, 6.10542686613735438204102033904, 6.92654382232964623444551127333, 7.04640739208492592152124389790, 7.57262719675420339795077380394, 8.176084068298713639108636445033, 8.607746552631514427123925111892, 8.910182480217836198499580298705, 9.222054175908024045832977661098, 9.505713386376568673210476808105, 10.55453481640419041740529738903

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