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Properties

Label 4-1520e2-1.1-c0e2-0-3
Degree $4$
Conductor $2310400$
Sign $1$
Analytic cond. $0.575441$
Root an. cond. $0.870964$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·19-s + 3·25-s + 2·49-s − 81-s − 4·95-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 2·5-s + 2·19-s + 3·25-s + 2·49-s − 81-s − 4·95-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2310400\)    =    \(2^{8} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.575441\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2310400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7559212634\)
\(L(\frac12)\) \(\approx\) \(0.7559212634\)
\(L(1)\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.881810391397916413009704096473, −9.333758392986555810566549022775, −8.909548534002982525343825109689, −8.768366917604205082708066046311, −8.142098538207784704121976491787, −7.75459383781529015979927766789, −7.53483326984986013544304140303, −7.26643416242760660844775979909, −6.69205321015119439143585820856, −6.42014215907750555193389890945, −5.48376300487681058471845699583, −5.40161300898695646180400397751, −4.84026935961743966237200129829, −4.25691348345654187211046544339, −3.93035808523073717032481274014, −3.53075132896040688641337643238, −2.92013503580586984101221815322, −2.64440395423594903169420657159, −1.47050220859465655908048266546, −0.76980421142416509575931543624, 0.76980421142416509575931543624, 1.47050220859465655908048266546, 2.64440395423594903169420657159, 2.92013503580586984101221815322, 3.53075132896040688641337643238, 3.93035808523073717032481274014, 4.25691348345654187211046544339, 4.84026935961743966237200129829, 5.40161300898695646180400397751, 5.48376300487681058471845699583, 6.42014215907750555193389890945, 6.69205321015119439143585820856, 7.26643416242760660844775979909, 7.53483326984986013544304140303, 7.75459383781529015979927766789, 8.142098538207784704121976491787, 8.768366917604205082708066046311, 8.909548534002982525343825109689, 9.333758392986555810566549022775, 9.881810391397916413009704096473

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