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Properties

Label 4-88e4-1.1-c1e2-0-31
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5-s − 7-s + 2·9-s + 7·13-s − 3·15-s − 7·17-s − 19-s + 3·21-s − 4·23-s − 8·25-s + 6·27-s + 15·29-s + 5·31-s − 35-s − 3·37-s − 21·39-s − 15·41-s + 2·45-s − 5·47-s − 2·49-s + 21·51-s − 3·53-s + 3·57-s − 9·59-s + 7·61-s − 2·63-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s − 0.377·7-s + 2/3·9-s + 1.94·13-s − 0.774·15-s − 1.69·17-s − 0.229·19-s + 0.654·21-s − 0.834·23-s − 8/5·25-s + 1.15·27-s + 2.78·29-s + 0.898·31-s − 0.169·35-s − 0.493·37-s − 3.36·39-s − 2.34·41-s + 0.298·45-s − 0.729·47-s − 2/7·49-s + 2.94·51-s − 0.412·53-s + 0.397·57-s − 1.17·59-s + 0.896·61-s − 0.251·63-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: \(2\)
Selberg data: \((4,\ 59969536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.3.d_h
5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_j
7$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_d
13$D_{4}$ \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.13.ah_bl
17$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bt
19$D_{4}$ \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_bb
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_be
29$D_{4}$ \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.29.ap_ej
31$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_cf
37$D_{4}$ \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_cn
41$D_{4}$ \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.41.p_fh
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_dv
53$D_{4}$ \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_cz
59$D_{4}$ \( 1 + 9 T + 107 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_ed
61$D_{4}$ \( 1 - 7 T + 133 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_fd
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$D_{4}$ \( 1 - 15 T + 197 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.71.ap_hp
73$D_{4}$ \( 1 - 13 T + 157 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.73.an_gb
79$D_{4}$ \( 1 + 21 T + 257 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.79.v_jx
83$D_{4}$ \( 1 - 3 T - 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_abr
89$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.89.i_gs
97$D_{4}$ \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_ej
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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

−7.79539216386094199302181449325, −6.99792583459978979798013861358, −6.62614464148682861655473124641, −6.58076219084268386812479722988, −6.24763085755931924077799331098, −6.17750933025796630155021666408, −5.63093883279579048672727657168, −5.36442401529314985391387367471, −4.86169112998878002778933928175, −4.73643223435239162356545661583, −4.05856307348521993061030632278, −3.97560852899432497302459976466, −3.16940242764768894050474435415, −3.14240798598708818813492875752, −2.15614208802078379400246129269, −2.15368105763101421397193769814, −1.29108347921860709782424458771, −0.973622721877319512335642925344, 0, 0, 0.973622721877319512335642925344, 1.29108347921860709782424458771, 2.15368105763101421397193769814, 2.15614208802078379400246129269, 3.14240798598708818813492875752, 3.16940242764768894050474435415, 3.97560852899432497302459976466, 4.05856307348521993061030632278, 4.73643223435239162356545661583, 4.86169112998878002778933928175, 5.36442401529314985391387367471, 5.63093883279579048672727657168, 6.17750933025796630155021666408, 6.24763085755931924077799331098, 6.58076219084268386812479722988, 6.62614464148682861655473124641, 6.99792583459978979798013861358, 7.79539216386094199302181449325

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