Properties

Label 4-882e2-1.1-c1e2-0-26
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-s − 2·5-s − 8-s − 2·10-s − 4·11-s + 12·13-s − 16-s + 2·17-s + 4·19-s − 4·22-s + 8·23-s + 5·25-s + 12·26-s + 4·29-s + 2·34-s + 10·37-s + 4·38-s + 2·40-s + 12·41-s − 8·43-s + 8·46-s + 5·50-s + 6·53-s + 8·55-s + 4·58-s + 4·59-s − 6·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.894·5-s − 0.353·8-s − 0.632·10-s − 1.20·11-s + 3.32·13-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s + 1.66·23-s + 25-s + 2.35·26-s + 0.742·29-s + 0.342·34-s + 1.64·37-s + 0.648·38-s + 0.316·40-s + 1.87·41-s − 1.21·43-s + 1.17·46-s + 0.707·50-s + 0.824·53-s + 1.07·55-s + 0.525·58-s + 0.520·59-s − 0.768·61-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

\(L(1)\) \(\approx\) \(2.960580091\)
\(L(\frac12)\) \(\approx\) \(2.960580091\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−10.41377193784944412020647326289, −10.14761129533523567393452524842, −9.314369857420179080449819758875, −9.064187811421767817867399268588, −8.475750797779161716523161779607, −8.382815734776921577968867096934, −7.82153758142460625344061714934, −7.41199538248759883085030891454, −6.91047494467057789490435692731, −6.35105789369861783269009979981, −5.83714246659870538106485144633, −5.64920998036338899743356264275, −5.02824940536341355363691394842, −4.37472688917563555672833112039, −4.16854941963872319445305655152, −3.38205691343224390955132470946, −3.14238621680038270067620355630, −2.70324186124067730048807065423, −1.29256973651247885409143718191, −0.899294285508509511139315567416, 0.899294285508509511139315567416, 1.29256973651247885409143718191, 2.70324186124067730048807065423, 3.14238621680038270067620355630, 3.38205691343224390955132470946, 4.16854941963872319445305655152, 4.37472688917563555672833112039, 5.02824940536341355363691394842, 5.64920998036338899743356264275, 5.83714246659870538106485144633, 6.35105789369861783269009979981, 6.91047494467057789490435692731, 7.41199538248759883085030891454, 7.82153758142460625344061714934, 8.382815734776921577968867096934, 8.475750797779161716523161779607, 9.064187811421767817867399268588, 9.314369857420179080449819758875, 10.14761129533523567393452524842, 10.41377193784944412020647326289

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