Properties

Label 4-442368-1.1-c1e2-0-37
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·7-s + 9-s + 12·13-s − 4·21-s − 10·25-s − 27-s + 20·31-s + 4·37-s − 12·39-s + 16·43-s − 2·49-s + 4·61-s + 4·63-s + 8·67-s − 20·73-s + 10·75-s − 12·79-s + 81-s + 48·91-s − 20·93-s − 12·97-s − 20·103-s − 4·109-s − 4·111-s + 12·117-s − 6·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s + 3.32·13-s − 0.872·21-s − 2·25-s − 0.192·27-s + 3.59·31-s + 0.657·37-s − 1.92·39-s + 2.43·43-s − 2/7·49-s + 0.512·61-s + 0.503·63-s + 0.977·67-s − 2.34·73-s + 1.15·75-s − 1.35·79-s + 1/9·81-s + 5.03·91-s − 2.07·93-s − 1.21·97-s − 1.97·103-s − 0.383·109-s − 0.379·111-s + 1.10·117-s − 0.545·121-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

\(L(1)\) \(\approx\) \(2.478326877\)
\(L(\frac12)\) \(\approx\) \(2.478326877\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.79.m_hm
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
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
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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

−8.396078230698006089481678801982, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −7.31199108027851415208504538208, −6.40139715881748782453956361829, −6.32971218305453945925176285036, −5.76550563678950678115455651188, −5.52928834758807112839354938677, −4.57667813854686016923499181308, −4.27440644248673667608699010194, −3.95147465503398880690094499828, −3.12662782741322079682173901720, −2.30530051971351607466902904067, −1.27109672072212401682622254955, −1.16901968292393942798169217496, 1.16901968292393942798169217496, 1.27109672072212401682622254955, 2.30530051971351607466902904067, 3.12662782741322079682173901720, 3.95147465503398880690094499828, 4.27440644248673667608699010194, 4.57667813854686016923499181308, 5.52928834758807112839354938677, 5.76550563678950678115455651188, 6.32971218305453945925176285036, 6.40139715881748782453956361829, 7.31199108027851415208504538208, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 8.396078230698006089481678801982

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