Properties

Label 4-24e4-1.1-c1e2-0-37
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
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  − 3·3-s + 6·5-s + 6·7-s + 6·9-s − 3·11-s + 4·13-s − 18·15-s − 18·21-s + 19·25-s − 9·27-s + 6·29-s + 9·33-s + 36·35-s − 4·37-s − 12·39-s − 9·41-s + 9·43-s + 36·45-s + 12·47-s + 17·49-s − 18·55-s − 15·59-s + 8·61-s + 36·63-s + 24·65-s − 15·67-s + 12·71-s + ⋯
L(s)  = 1  − 1.73·3-s + 2.68·5-s + 2.26·7-s + 2·9-s − 0.904·11-s + 1.10·13-s − 4.64·15-s − 3.92·21-s + 19/5·25-s − 1.73·27-s + 1.11·29-s + 1.56·33-s + 6.08·35-s − 0.657·37-s − 1.92·39-s − 1.40·41-s + 1.37·43-s + 5.36·45-s + 1.75·47-s + 17/7·49-s − 2.42·55-s − 1.95·59-s + 1.02·61-s + 4.53·63-s + 2.97·65-s − 1.83·67-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.474381999\)
\(L(\frac12)\) \(\approx\) \(2.474381999\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
good5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.5.ag_r
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.7.ag_t
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.17.a_abf
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_bp
31$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.31.a_bf
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_cq
43$C_2^2$ \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.43.aj_cs
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_dt
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2^2$ \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_gk
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_d
67$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_fm
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_dn
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.89.a_o
97$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.97.n_cu
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

−10.74058731024643857336293671357, −10.63649720319680249865430626686, −10.34186580140632018668047736394, −9.867603382615333971772600237512, −9.184223038330498079473009340110, −8.957725836169782290548467459371, −8.100971969584353498748225622445, −8.074436506664780633222112159620, −6.96075759571043762402783946001, −6.93691878272699342381537590660, −5.96993288783695544809878473419, −5.91991095027876157514805336914, −5.46300293828809589109798412779, −5.18402986342618963934834912140, −4.61793509258135293692749458036, −4.25963847172732980825448104873, −2.87341808811382447272501028659, −2.13359376499004227954664019551, −1.50231020726581931292640182387, −1.22369546718801342396975393974, 1.22369546718801342396975393974, 1.50231020726581931292640182387, 2.13359376499004227954664019551, 2.87341808811382447272501028659, 4.25963847172732980825448104873, 4.61793509258135293692749458036, 5.18402986342618963934834912140, 5.46300293828809589109798412779, 5.91991095027876157514805336914, 5.96993288783695544809878473419, 6.93691878272699342381537590660, 6.96075759571043762402783946001, 8.074436506664780633222112159620, 8.100971969584353498748225622445, 8.957725836169782290548467459371, 9.184223038330498079473009340110, 9.867603382615333971772600237512, 10.34186580140632018668047736394, 10.63649720319680249865430626686, 10.74058731024643857336293671357

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