Properties

Label 4-442368-1.1-c1e2-0-38
Degree $4$
Conductor $442368$
Sign $1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
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 + 9-s + 8·11-s + 12·13-s − 8·23-s − 10·25-s + 27-s + 8·33-s + 4·37-s + 12·39-s + 24·47-s − 10·49-s + 8·59-s + 4·61-s − 8·69-s + 8·71-s − 20·73-s − 10·75-s + 81-s − 24·83-s − 12·97-s + 8·99-s + 24·107-s − 4·109-s + 4·111-s + 12·117-s + 26·121-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 2.41·11-s + 3.32·13-s − 1.66·23-s − 2·25-s + 0.192·27-s + 1.39·33-s + 0.657·37-s + 1.92·39-s + 3.50·47-s − 1.42·49-s + 1.04·59-s + 0.512·61-s − 0.963·69-s + 0.949·71-s − 2.34·73-s − 1.15·75-s + 1/9·81-s − 2.63·83-s − 1.21·97-s + 0.804·99-s + 2.32·107-s − 0.383·109-s + 0.379·111-s + 1.10·117-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.321867930\)
\(L(\frac12)\) \(\approx\) \(3.321867930\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.79.a_es
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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

−8.658903060264415544055202217864, −8.396078230698006089481678801982, −7.70036027517747780731408099423, −7.31199108027851415208504538208, −6.52243750033071865852983380426, −6.32971218305453945925176285036, −5.76550563678950678115455651188, −5.75568326693151721019291253980, −4.27440644248673667608699010194, −3.95147465503398880690094499828, −3.94434944725762226728078180666, −3.36330701716196318281397292070, −2.30530051971351607466902904067, −1.50413098873776121963751941726, −1.16901968292393942798169217496, 1.16901968292393942798169217496, 1.50413098873776121963751941726, 2.30530051971351607466902904067, 3.36330701716196318281397292070, 3.94434944725762226728078180666, 3.95147465503398880690094499828, 4.27440644248673667608699010194, 5.75568326693151721019291253980, 5.76550563678950678115455651188, 6.32971218305453945925176285036, 6.52243750033071865852983380426, 7.31199108027851415208504538208, 7.70036027517747780731408099423, 8.396078230698006089481678801982, 8.658903060264415544055202217864

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