Properties

Label 4-5586e2-1.1-c1e2-0-11
Degree $4$
Conductor $31203396$
Sign $1$
Analytic cond. $1989.55$
Root an. cond. $6.67865$
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 − 2·3-s + 3·4-s − 6·5-s − 4·6-s + 4·8-s + 3·9-s − 12·10-s − 2·11-s − 6·12-s + 12·15-s + 5·16-s + 6·18-s + 2·19-s − 18·20-s − 4·22-s + 4·23-s − 8·24-s + 19·25-s − 4·27-s + 10·29-s + 24·30-s − 10·31-s + 6·32-s + 4·33-s + 9·36-s − 8·37-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 2.68·5-s − 1.63·6-s + 1.41·8-s + 9-s − 3.79·10-s − 0.603·11-s − 1.73·12-s + 3.09·15-s + 5/4·16-s + 1.41·18-s + 0.458·19-s − 4.02·20-s − 0.852·22-s + 0.834·23-s − 1.63·24-s + 19/5·25-s − 0.769·27-s + 1.85·29-s + 4.38·30-s − 1.79·31-s + 1.06·32-s + 0.696·33-s + 3/2·36-s − 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31203396\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1989.55\)
Root analytic conductor: \(6.67865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5586} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 31203396,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 55 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 24 T + 282 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 159 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 14 T + 241 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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

−7.69455050728402810596011300212, −7.38556353336312878038613447465, −7.20376195431669540433880207836, −7.07234083213947470372541896923, −6.47227086400524367552556999968, −6.15748150583024484928937796812, −5.67123041018279430921433965550, −5.29816877415669655321696242430, −4.89562429515684703647924779871, −4.80805777510636754975723185551, −4.14077065403950660365487494320, −4.13543208222462487273065440528, −3.50183375947912428865665553050, −3.42342812643037481036938482380, −2.73363027048482320977641619538, −2.48467441877462232304901553534, −1.35857257526470938939943373715, −1.14098873665669935040871863935, 0, 0, 1.14098873665669935040871863935, 1.35857257526470938939943373715, 2.48467441877462232304901553534, 2.73363027048482320977641619538, 3.42342812643037481036938482380, 3.50183375947912428865665553050, 4.13543208222462487273065440528, 4.14077065403950660365487494320, 4.80805777510636754975723185551, 4.89562429515684703647924779871, 5.29816877415669655321696242430, 5.67123041018279430921433965550, 6.15748150583024484928937796812, 6.47227086400524367552556999968, 7.07234083213947470372541896923, 7.20376195431669540433880207836, 7.38556353336312878038613447465, 7.69455050728402810596011300212

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