Properties

Label 4-28e4-1.1-c1e2-0-13
Degree $4$
Conductor $614656$
Sign $1$
Analytic cond. $39.1909$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s + 3·9-s − 3·11-s − 4·13-s − 3·15-s + 3·17-s + 19-s + 3·23-s + 5·25-s − 8·27-s − 12·29-s + 7·31-s + 3·33-s + 37-s + 4·39-s − 12·41-s + 8·43-s + 9·45-s + 9·47-s − 3·51-s − 3·53-s − 9·55-s − 57-s − 9·59-s − 61-s − 12·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 9-s − 0.904·11-s − 1.10·13-s − 0.774·15-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 25-s − 1.53·27-s − 2.22·29-s + 1.25·31-s + 0.522·33-s + 0.164·37-s + 0.640·39-s − 1.87·41-s + 1.21·43-s + 1.34·45-s + 1.31·47-s − 0.420·51-s − 0.412·53-s − 1.21·55-s − 0.132·57-s − 1.17·59-s − 0.128·61-s − 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(39.1909\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842263128\)
\(L(\frac12)\) \(\approx\) \(1.842263128\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.44935051711723392699614595410, −9.916376550706979719229660705969, −9.909077189914488117649018297796, −9.293682340537926849137024674493, −9.094111790654333707311549372934, −8.408598150886657313185392215239, −7.62926231066195278646976716819, −7.38405865827004440009669189840, −7.32843964470902179094384314182, −6.36785745266162731248624384606, −6.12400464123138238829553209446, −5.56479281380301108940431307860, −5.27620547862612895005068209007, −4.78576284752321927715712470940, −4.34383767953976322852569964045, −3.44333869827717705299929632929, −2.96698847671378968283122958177, −2.01699466734284284737264425041, −1.88021410095585668506948943359, −0.70481772857522623718993770402, 0.70481772857522623718993770402, 1.88021410095585668506948943359, 2.01699466734284284737264425041, 2.96698847671378968283122958177, 3.44333869827717705299929632929, 4.34383767953976322852569964045, 4.78576284752321927715712470940, 5.27620547862612895005068209007, 5.56479281380301108940431307860, 6.12400464123138238829553209446, 6.36785745266162731248624384606, 7.32843964470902179094384314182, 7.38405865827004440009669189840, 7.62926231066195278646976716819, 8.408598150886657313185392215239, 9.094111790654333707311549372934, 9.293682340537926849137024674493, 9.909077189914488117649018297796, 9.916376550706979719229660705969, 10.44935051711723392699614595410

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