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Properties

Label 4-117e2-1.1-c1e2-0-13
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $0.872822$
Root an. cond. $0.966565$
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  + 4-s + 3·5-s + 3·7-s − 3·9-s − 2·13-s − 3·16-s − 3·17-s − 3·19-s + 3·20-s + 3·23-s + 25-s + 3·28-s − 12·29-s + 15·31-s + 9·35-s − 3·36-s − 9·37-s − 21·41-s + 43-s − 9·45-s + 9·47-s − 49-s − 2·52-s + 12·53-s + 5·61-s − 9·63-s − 7·64-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.34·5-s + 1.13·7-s − 9-s − 0.554·13-s − 3/4·16-s − 0.727·17-s − 0.688·19-s + 0.670·20-s + 0.625·23-s + 1/5·25-s + 0.566·28-s − 2.22·29-s + 2.69·31-s + 1.52·35-s − 1/2·36-s − 1.47·37-s − 3.27·41-s + 0.152·43-s − 1.34·45-s + 1.31·47-s − 1/7·49-s − 0.277·52-s + 1.64·53-s + 0.640·61-s − 1.13·63-s − 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.872822\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.398065909\)
\(L(\frac12)\) \(\approx\) \(1.398065909\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_i
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ad_k
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_w
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.ap_ec
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.j_cm
41$C_2^2$ \( 1 + 21 T + 188 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.41.v_hg
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_cw
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.59.a_aec
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_abk
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.av_ig
71$C_2^2$ \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.71.p_fq
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.73.a_adu
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.79.al_bq
83$C_2^2$ \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_eg
89$C_2^2$ \( 1 - 27 T + 332 T^{2} - 27 p T^{3} + p^{2} T^{4} \) 2.89.abb_mu
97$C_2^2$ \( 1 - 27 T + 340 T^{2} - 27 p T^{3} + p^{2} T^{4} \) 2.97.abb_nc
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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

−13.66759014459622925712490442218, −13.51607083454318894937519297152, −12.96275518142648054401127533643, −11.87309586014801198716934559275, −11.80430470715270512324443966518, −11.32979864950134251443571025590, −10.45909693767849444608675620981, −10.42249421526834460621705263113, −9.555554372129609893289075813831, −8.880287183576737153781596149957, −8.608430982151890089595969902945, −7.902841901973221556917586273794, −7.07519335918192037213891224748, −6.54227041416560991394916286023, −5.97787034747356742681129252949, −5.07752838344390214932622301790, −4.93825816823722969879644120524, −3.61107539215575108044696718396, −2.27075375258071963491569305690, −2.05794273235526551664889387787, 2.05794273235526551664889387787, 2.27075375258071963491569305690, 3.61107539215575108044696718396, 4.93825816823722969879644120524, 5.07752838344390214932622301790, 5.97787034747356742681129252949, 6.54227041416560991394916286023, 7.07519335918192037213891224748, 7.902841901973221556917586273794, 8.608430982151890089595969902945, 8.880287183576737153781596149957, 9.555554372129609893289075813831, 10.42249421526834460621705263113, 10.45909693767849444608675620981, 11.32979864950134251443571025590, 11.80430470715270512324443966518, 11.87309586014801198716934559275, 12.96275518142648054401127533643, 13.51607083454318894937519297152, 13.66759014459622925712490442218

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