Properties

Label 4-51e4-1.1-c1e2-0-3
Degree $4$
Conductor $6765201$
Sign $1$
Analytic cond. $431.355$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s − 6·7-s + 4·11-s + 2·13-s + 12·14-s + 16-s − 6·19-s − 8·22-s − 2·25-s − 4·26-s − 6·28-s − 12·29-s − 2·31-s + 2·32-s − 2·37-s + 12·38-s + 12·41-s + 6·43-s + 4·44-s + 8·47-s + 13·49-s + 4·50-s + 2·52-s − 8·53-s + 24·58-s + 12·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 2.26·7-s + 1.20·11-s + 0.554·13-s + 3.20·14-s + 1/4·16-s − 1.37·19-s − 1.70·22-s − 2/5·25-s − 0.784·26-s − 1.13·28-s − 2.22·29-s − 0.359·31-s + 0.353·32-s − 0.328·37-s + 1.94·38-s + 1.87·41-s + 0.914·43-s + 0.603·44-s + 1.16·47-s + 13/7·49-s + 0.565·50-s + 0.277·52-s − 1.09·53-s + 3.15·58-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6765201\)    =    \(3^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(431.355\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6765201,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
17 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T + 91 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 187 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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

−9.028303091962508915209678269616, −8.609857882850148437990252060276, −7.927859954182017115057084597460, −7.64023381984453899902917846840, −7.24997110213363823601893646004, −6.74929136559912823282056237142, −6.48999952999554968796660487055, −6.06230832795072243431084000379, −5.84632618014579725384077394860, −5.49459510061699868608810406753, −4.50232029794972677023856294930, −4.14257966206617858258723237874, −3.70982402960184216360753589386, −3.54105021443961775501488211801, −2.74714204635400215149953393905, −2.42532790168208101583075808002, −1.58871635989500288686878782469, −1.01655854163611501623981731142, 0, 0, 1.01655854163611501623981731142, 1.58871635989500288686878782469, 2.42532790168208101583075808002, 2.74714204635400215149953393905, 3.54105021443961775501488211801, 3.70982402960184216360753589386, 4.14257966206617858258723237874, 4.50232029794972677023856294930, 5.49459510061699868608810406753, 5.84632618014579725384077394860, 6.06230832795072243431084000379, 6.48999952999554968796660487055, 6.74929136559912823282056237142, 7.24997110213363823601893646004, 7.64023381984453899902917846840, 7.927859954182017115057084597460, 8.609857882850148437990252060276, 9.028303091962508915209678269616

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