Properties

Label 4-882e2-1.1-c1e2-0-18
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $49.6011$
Root an. cond. $2.65382$
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  − 2·2-s + 3·3-s + 3·4-s − 3·5-s − 6·6-s − 4·8-s + 6·9-s + 6·10-s + 6·11-s + 9·12-s + 2·13-s − 9·15-s + 5·16-s + 6·17-s − 12·18-s − 7·19-s − 9·20-s − 12·22-s − 3·23-s − 12·24-s + 5·25-s − 4·26-s + 9·27-s − 6·29-s + 18·30-s − 4·31-s − 6·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 3/2·4-s − 1.34·5-s − 2.44·6-s − 1.41·8-s + 2·9-s + 1.89·10-s + 1.80·11-s + 2.59·12-s + 0.554·13-s − 2.32·15-s + 5/4·16-s + 1.45·17-s − 2.82·18-s − 1.60·19-s − 2.01·20-s − 2.55·22-s − 0.625·23-s − 2.44·24-s + 25-s − 0.784·26-s + 1.73·27-s − 1.11·29-s + 3.28·30-s − 0.718·31-s − 1.06·32-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(49.6011\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.683159637\)
\(L(\frac12)\) \(\approx\) \(1.683159637\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−10.30342881602350466629626547855, −9.580943614488192436286195946268, −9.378211750079020802052420405131, −9.085960523225293808189055890237, −8.635880852005952218943819256235, −8.283882833156454088544932815282, −7.87444730208417311617148945876, −7.76210689350065334241214993074, −7.20001285286957715535749432011, −6.68607737276687456952568019909, −6.43306271838766420082089731189, −5.78024673367671909930095944143, −4.89829546917896514722514151712, −3.99602832788807490235429905619, −3.89318796247202082018883405276, −3.48347600463853945544674083816, −2.95957098769143775327443412310, −1.90123015979025708737691624133, −1.75456471170968396787413712455, −0.73356393598858232553921223617, 0.73356393598858232553921223617, 1.75456471170968396787413712455, 1.90123015979025708737691624133, 2.95957098769143775327443412310, 3.48347600463853945544674083816, 3.89318796247202082018883405276, 3.99602832788807490235429905619, 4.89829546917896514722514151712, 5.78024673367671909930095944143, 6.43306271838766420082089731189, 6.68607737276687456952568019909, 7.20001285286957715535749432011, 7.76210689350065334241214993074, 7.87444730208417311617148945876, 8.283882833156454088544932815282, 8.635880852005952218943819256235, 9.085960523225293808189055890237, 9.378211750079020802052420405131, 9.580943614488192436286195946268, 10.30342881602350466629626547855

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