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Properties

Label 4-2184e2-1.1-c1e2-0-0
Degree $4$
Conductor $4769856$
Sign $1$
Analytic cond. $304.130$
Root an. cond. $4.17604$
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 − 4·5-s + 7-s − 3·11-s + 5·13-s − 4·15-s + 17-s − 5·19-s + 21-s − 2·23-s + 2·25-s − 27-s − 29-s − 4·31-s − 3·33-s − 4·35-s − 2·37-s + 5·39-s − 5·41-s − 2·43-s − 6·47-s + 51-s − 18·53-s + 12·55-s − 5·57-s + 6·59-s − 13·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 0.377·7-s − 0.904·11-s + 1.38·13-s − 1.03·15-s + 0.242·17-s − 1.14·19-s + 0.218·21-s − 0.417·23-s + 2/5·25-s − 0.192·27-s − 0.185·29-s − 0.718·31-s − 0.522·33-s − 0.676·35-s − 0.328·37-s + 0.800·39-s − 0.780·41-s − 0.304·43-s − 0.875·47-s + 0.140·51-s − 2.47·53-s + 1.61·55-s − 0.662·57-s + 0.781·59-s − 1.66·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4769856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4769856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4769856\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(304.130\)
Root analytic conductor: \(4.17604\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4769856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01164799893\)
\(L(\frac12)\) \(\approx\) \(0.01164799893\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
17$C_2^2$ \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_aq
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_at
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_abc
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_aq
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_abn
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.n_ee
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_cv
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.73.q_ic
79$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.79.bi_rf
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 5 T - 64 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.89.af_acm
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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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

−9.227789365737865995697925123338, −8.568653312693738708545312248012, −8.380009880467043728357735390085, −8.120343403412617923581997239299, −7.927847034534505102612872080206, −7.38786032340618359287146298779, −7.21116622257529093826410447669, −6.39016575840919294836552564822, −6.35774447924758112027462810385, −5.53612321073375945555934117878, −5.41279480621732909387282247274, −4.51933285665320538382649467693, −4.42779789062125074069070629527, −3.86747194683515132857045520071, −3.57107461380141079247859925308, −3.09060196927767617518880475427, −2.66930837093821202355169454721, −1.70702843108980324869324587382, −1.52263685450389964913780832738, −0.03542177782556320914266792734, 0.03542177782556320914266792734, 1.52263685450389964913780832738, 1.70702843108980324869324587382, 2.66930837093821202355169454721, 3.09060196927767617518880475427, 3.57107461380141079247859925308, 3.86747194683515132857045520071, 4.42779789062125074069070629527, 4.51933285665320538382649467693, 5.41279480621732909387282247274, 5.53612321073375945555934117878, 6.35774447924758112027462810385, 6.39016575840919294836552564822, 7.21116622257529093826410447669, 7.38786032340618359287146298779, 7.927847034534505102612872080206, 8.120343403412617923581997239299, 8.380009880467043728357735390085, 8.568653312693738708545312248012, 9.227789365737865995697925123338

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