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Properties

Label 4-363e2-1.1-c1e2-0-18
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 2·4-s − 3·5-s − 2·6-s − 6·7-s + 3·8-s + 3·9-s + 3·10-s − 4·12-s − 8·13-s + 6·14-s − 6·15-s + 16-s − 17-s − 3·18-s − 5·19-s + 6·20-s − 12·21-s − 2·23-s + 6·24-s − 2·25-s + 8·26-s + 4·27-s + 12·28-s + 6·30-s − 31-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 4-s − 1.34·5-s − 0.816·6-s − 2.26·7-s + 1.06·8-s + 9-s + 0.948·10-s − 1.15·12-s − 2.21·13-s + 1.60·14-s − 1.54·15-s + 1/4·16-s − 0.242·17-s − 0.707·18-s − 1.14·19-s + 1.34·20-s − 2.61·21-s − 0.417·23-s + 1.22·24-s − 2/5·25-s + 1.56·26-s + 0.769·27-s + 2.26·28-s + 1.09·30-s − 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_d
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_l
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bl
17$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) 2.17.b_bh
19$D_{4}$ \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_bh
23$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bb
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$D_{4}$ \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_bz
37$D_{4}$ \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cv
41$D_{4}$ \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_l
43$D_{4}$ \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_dt
47$D_{4}$ \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) 2.47.ab_dp
53$D_{4}$ \( 1 + 17 T + 177 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.53.r_gv
59$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.59.af_cl
61$D_{4}$ \( 1 + 9 T + 131 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.61.j_fb
67$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_bh
71$D_{4}$ \( 1 - 9 T + 61 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_cj
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_fm
79$D_{4}$ \( 1 + 10 T + 163 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_gh
83$D_{4}$ \( 1 - 12 T + 157 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_gb
89$D_{4}$ \( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_hb
97$D_{4}$ \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ar
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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

−11.03365687565380218311807407594, −10.44962004892279487113516353833, −9.853554053915169086297688218088, −9.768570950460357282081378252608, −9.368161890918320204769588635155, −9.072922961471080520082514363765, −8.262328614028359519534830216125, −8.193649732012210798534917023753, −7.50470918349601455325913291569, −7.23902800568546240059971405639, −6.56497603865813640787210971213, −6.12188962970225857637937319661, −4.93286077599260726790850476019, −4.61258388867120864853280197895, −3.86764959082198904621871401652, −3.58014662113298503108212757707, −2.88545461561253685689728106456, −2.15407132052680465264893855070, 0, 0, 2.15407132052680465264893855070, 2.88545461561253685689728106456, 3.58014662113298503108212757707, 3.86764959082198904621871401652, 4.61258388867120864853280197895, 4.93286077599260726790850476019, 6.12188962970225857637937319661, 6.56497603865813640787210971213, 7.23902800568546240059971405639, 7.50470918349601455325913291569, 8.193649732012210798534917023753, 8.262328614028359519534830216125, 9.072922961471080520082514363765, 9.368161890918320204769588635155, 9.768570950460357282081378252608, 9.853554053915169086297688218088, 10.44962004892279487113516353833, 11.03365687565380218311807407594

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