Properties

Label 4-88e4-1.1-c1e2-0-24
Degree $4$
Conductor $59969536$
Sign $1$
Analytic cond. $3823.70$
Root an. cond. $7.86359$
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  + 2·3-s − 2·7-s + 2·9-s + 8·13-s − 2·17-s + 2·19-s − 4·21-s + 6·23-s − 5·25-s + 6·27-s + 8·29-s + 14·31-s − 4·37-s + 16·39-s − 6·41-s + 12·43-s − 2·47-s − 6·49-s − 4·51-s + 4·53-s + 4·57-s + 20·59-s + 12·61-s − 4·63-s + 6·67-s + 12·69-s − 8·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 2/3·9-s + 2.21·13-s − 0.485·17-s + 0.458·19-s − 0.872·21-s + 1.25·23-s − 25-s + 1.15·27-s + 1.48·29-s + 2.51·31-s − 0.657·37-s + 2.56·39-s − 0.937·41-s + 1.82·43-s − 0.291·47-s − 6/7·49-s − 0.560·51-s + 0.549·53-s + 0.529·57-s + 2.60·59-s + 1.53·61-s − 0.503·63-s + 0.733·67-s + 1.44·69-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(59969536\)    =    \(2^{12} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3823.70\)
Root analytic conductor: \(7.86359\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 59969536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.208742736\)
\(L(\frac12)\) \(\approx\) \(7.208742736\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.5.a_f
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_k
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bl
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.17.c_bj
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ag
23$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_by
29$D_{4}$ \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_cr
31$C_4$ \( 1 - 14 T + 106 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.31.ao_ec
37$D_{4}$ \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cv
41$D_{4}$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_ct
43$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_dy
47$D_{4}$ \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_by
53$D_{4}$ \( 1 - 4 T - 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_ap
59$D_{4}$ \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.59.au_hq
61$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_da
67$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_fi
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_hi
79$D_{4}$ \( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_gw
83$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_bq
89$D_{4}$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_ab
97$D_{4}$ \( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_et
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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

−8.044471243572849677857917646576, −8.022205704752431972832460160095, −7.22453140178347843168139748835, −6.88172924556297882195205984237, −6.81562208889035677288743113307, −6.36995823172612137378367702183, −5.99237880602037395093867455142, −5.77934096647051183105337423823, −5.19602439309640520581239399942, −4.62885384147634618485194409480, −4.58929046993053267333466596191, −3.87196151233621846447812410388, −3.65718719957586659978314061460, −3.36918976124577144868678988684, −2.76339330711462844934859266406, −2.73707654294940986815809538155, −2.16824523886882276588501431657, −1.49188805887735545414826466519, −0.919492837690981182666783255991, −0.75730568063537728737024391841, 0.75730568063537728737024391841, 0.919492837690981182666783255991, 1.49188805887735545414826466519, 2.16824523886882276588501431657, 2.73707654294940986815809538155, 2.76339330711462844934859266406, 3.36918976124577144868678988684, 3.65718719957586659978314061460, 3.87196151233621846447812410388, 4.58929046993053267333466596191, 4.62885384147634618485194409480, 5.19602439309640520581239399942, 5.77934096647051183105337423823, 5.99237880602037395093867455142, 6.36995823172612137378367702183, 6.81562208889035677288743113307, 6.88172924556297882195205984237, 7.22453140178347843168139748835, 8.022205704752431972832460160095, 8.044471243572849677857917646576

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