Properties

Label 4-1492992-1.1-c1e2-0-10
Degree $4$
Conductor $1492992$
Sign $1$
Analytic cond. $95.1944$
Root an. cond. $3.12358$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·11-s + 4·17-s + 6·19-s − 6·25-s − 16·41-s + 24·43-s − 5·49-s + 12·59-s + 6·67-s − 30·73-s − 24·83-s + 20·89-s + 18·97-s − 12·107-s − 28·113-s + 86·121-s + ⋯
L(s)  = 1  + 3.61·11-s + 0.970·17-s + 1.37·19-s − 6/5·25-s − 2.49·41-s + 3.65·43-s − 5/7·49-s + 1.56·59-s + 0.733·67-s − 3.51·73-s − 2.63·83-s + 2.11·89-s + 1.82·97-s − 1.16·107-s − 2.63·113-s + 7.81·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1492992\)    =    \(2^{11} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(95.1944\)
Root analytic conductor: \(3.12358\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1492992,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.318341860\)
\(L(\frac12)\) \(\approx\) \(3.318341860\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.a_r
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.a_z
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.41.q_fq
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.a_er
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.67.ag_fn
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.73.be_oh
79$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.79.a_cz
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.97.as_kp
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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

−7.78639326445418573570183285345, −7.42468008636998742776243725148, −6.88359362105880509746395731542, −6.84703029813025683919349357796, −5.96035170554403781150034405946, −5.95444690232608248894153675717, −5.45584155430329122752354767088, −4.62403138193129190420684686156, −4.24643980416916525601100537552, −3.73252772900173333221011269732, −3.53215777188464305495336430815, −2.88076148445155716360589692436, −1.89419396016353367846966262393, −1.36990894878106263941737695602, −0.920256988552060674036203635836, 0.920256988552060674036203635836, 1.36990894878106263941737695602, 1.89419396016353367846966262393, 2.88076148445155716360589692436, 3.53215777188464305495336430815, 3.73252772900173333221011269732, 4.24643980416916525601100537552, 4.62403138193129190420684686156, 5.45584155430329122752354767088, 5.95444690232608248894153675717, 5.96035170554403781150034405946, 6.84703029813025683919349357796, 6.88359362105880509746395731542, 7.42468008636998742776243725148, 7.78639326445418573570183285345

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