Properties

Label 4-1710e2-1.1-c1e2-0-2
Degree $4$
Conductor $2924100$
Sign $1$
Analytic cond. $186.443$
Root an. cond. $3.69518$
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  + 2-s − 5-s − 2·7-s − 8-s − 10-s − 4·11-s + 3·13-s − 2·14-s − 16-s + 4·17-s + 8·19-s − 4·22-s − 6·23-s + 3·26-s − 10·29-s + 2·31-s + 4·34-s + 2·35-s − 10·37-s + 8·38-s + 40-s + 2·41-s + 5·43-s − 6·46-s − 11·49-s − 12·53-s + 4·55-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.832·13-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.852·22-s − 1.25·23-s + 0.588·26-s − 1.85·29-s + 0.359·31-s + 0.685·34-s + 0.338·35-s − 1.64·37-s + 1.29·38-s + 0.158·40-s + 0.312·41-s + 0.762·43-s − 0.884·46-s − 1.57·49-s − 1.64·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2924100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(186.443\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1710} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2924100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.299704334\)
\(L(\frac12)\) \(\approx\) \(1.299704334\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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

−9.521019738652887076679875482977, −9.263298283194245330597136393054, −8.735609011327007488369907528226, −8.248456154312129889804315922196, −7.79384514250086676435846351132, −7.52902298091379398069279549280, −7.39346916233407878148984357346, −6.54461899311020367173863015587, −6.26328222549425537948798407558, −5.68422017708166314169812374662, −5.61137450953318509240682647182, −4.93748198547746914391924650948, −4.76674731270656548056632891616, −3.82786039642966335783651288815, −3.69220654211477199101660299721, −3.16333239820633422659701529038, −2.97854462964633525383075663435, −2.04172403616598021533546531893, −1.43265587296690690164786913580, −0.38710904359593117618259384021, 0.38710904359593117618259384021, 1.43265587296690690164786913580, 2.04172403616598021533546531893, 2.97854462964633525383075663435, 3.16333239820633422659701529038, 3.69220654211477199101660299721, 3.82786039642966335783651288815, 4.76674731270656548056632891616, 4.93748198547746914391924650948, 5.61137450953318509240682647182, 5.68422017708166314169812374662, 6.26328222549425537948798407558, 6.54461899311020367173863015587, 7.39346916233407878148984357346, 7.52902298091379398069279549280, 7.79384514250086676435846351132, 8.248456154312129889804315922196, 8.735609011327007488369907528226, 9.263298283194245330597136393054, 9.521019738652887076679875482977

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