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Properties

Label 4-2520e2-1.1-c1e2-0-13
Degree $4$
Conductor $6350400$
Sign $1$
Analytic cond. $404.907$
Root an. cond. $4.48578$
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  − 5-s + 4·7-s − 11-s − 6·13-s − 2·17-s + 5·19-s + 7·23-s + 12·29-s − 4·31-s − 4·35-s + 5·37-s + 10·41-s + 12·43-s − 9·47-s + 9·49-s + 11·53-s + 55-s + 8·59-s + 12·61-s + 6·65-s + 4·67-s + 8·71-s − 12·73-s − 4·77-s − 14·79-s + 8·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 0.301·11-s − 1.66·13-s − 0.485·17-s + 1.14·19-s + 1.45·23-s + 2.22·29-s − 0.718·31-s − 0.676·35-s + 0.821·37-s + 1.56·41-s + 1.82·43-s − 1.31·47-s + 9/7·49-s + 1.51·53-s + 0.134·55-s + 1.04·59-s + 1.53·61-s + 0.744·65-s + 0.488·67-s + 0.949·71-s − 1.40·73-s − 0.455·77-s − 1.57·79-s + 0.878·83-s + 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6350400\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(404.907\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6350400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.236586461\)
\(L(\frac12)\) \(\approx\) \(3.236586461\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_ak
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_an
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_ba
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2^2$ \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.37.af_am
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.41.ak_ed
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_bi
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.53.al_cq
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_f
61$C_2^2$ \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_df
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_ct
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_en
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_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

−8.972297904479883238945647392118, −8.791946621318299684379746035300, −8.119280721857280814751684692802, −8.068892527288288142877442889650, −7.44288011746222660204987632086, −7.42581049719476043395770205293, −6.85275071721364271345223966334, −6.67278328980863737766427790831, −5.72948531995675413185595193614, −5.60254243951814978616321989377, −4.98385438459882192804490140952, −4.89492322202028181960637548424, −4.28800219002226862995000045631, −4.20051254021827887462923809206, −3.30838618782954384664330983332, −2.79531972754926024125097723619, −2.44757560121531930078913341989, −2.00001611572601599166939260611, −0.980643232192021474666304123604, −0.77128339469304737573022260051, 0.77128339469304737573022260051, 0.980643232192021474666304123604, 2.00001611572601599166939260611, 2.44757560121531930078913341989, 2.79531972754926024125097723619, 3.30838618782954384664330983332, 4.20051254021827887462923809206, 4.28800219002226862995000045631, 4.89492322202028181960637548424, 4.98385438459882192804490140952, 5.60254243951814978616321989377, 5.72948531995675413185595193614, 6.67278328980863737766427790831, 6.85275071721364271345223966334, 7.42581049719476043395770205293, 7.44288011746222660204987632086, 8.068892527288288142877442889650, 8.119280721857280814751684692802, 8.791946621318299684379746035300, 8.972297904479883238945647392118

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