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Properties

Label 4-3654e2-1.1-c1e2-0-11
Degree $4$
Conductor $13351716$
Sign $1$
Analytic cond. $851.316$
Root an. cond. $5.40160$
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  − 2·2-s + 3·4-s − 4·5-s + 2·7-s − 4·8-s + 8·10-s + 11-s + 3·13-s − 4·14-s + 5·16-s − 4·17-s + 3·19-s − 12·20-s − 2·22-s − 3·23-s + 2·25-s − 6·26-s + 6·28-s − 2·29-s − 4·31-s − 6·32-s + 8·34-s − 8·35-s + 3·37-s − 6·38-s + 16·40-s − 6·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.78·5-s + 0.755·7-s − 1.41·8-s + 2.52·10-s + 0.301·11-s + 0.832·13-s − 1.06·14-s + 5/4·16-s − 0.970·17-s + 0.688·19-s − 2.68·20-s − 0.426·22-s − 0.625·23-s + 2/5·25-s − 1.17·26-s + 1.13·28-s − 0.371·29-s − 0.718·31-s − 1.06·32-s + 1.37·34-s − 1.35·35-s + 0.493·37-s − 0.973·38-s + 2.52·40-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13351716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13351716 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13351716\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(851.316\)
Root analytic conductor: \(5.40160\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13351716,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_s
13$D_{4}$ \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_y
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$D_{4}$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_c
23$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_k
31$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_ac
37$D_{4}$ \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_cu
41$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cw
43$D_{4}$ \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_cs
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_ck
53$D_{4}$ \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \) 2.53.b_cq
59$D_{4}$ \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_de
61$D_{4}$ \( 1 - 22 T + 226 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.61.aw_is
67$D_{4}$ \( 1 + 15 T + 186 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_he
71$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_da
73$D_{4}$ \( 1 - 7 T + 120 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.73.ah_eq
79$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_ew
83$D_{4}$ \( 1 - 11 T + 90 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.83.al_dm
89$D_{4}$ \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.89.w_kw
97$D_{4}$ \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.97.aj_ee
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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

−8.227839302657972865682218318146, −8.144214142345946146944784820603, −7.74309430036937936417494119079, −7.33449156432664284571101526990, −7.16417015302239535967104674234, −6.72654241750902818200843694655, −6.21569155671181032516734739000, −5.95607206450521569803776821254, −5.26095683055226901473017944884, −5.06888163991834312506339145999, −4.27722418500781214572824406804, −4.04202540316800303734456461956, −3.61054123261623889624678982531, −3.40684771147111673326485675938, −2.41118858544775018862521200770, −2.31827224799800900953116068903, −1.30320477199035875353320980268, −1.26158846800231853041360675265, 0, 0, 1.26158846800231853041360675265, 1.30320477199035875353320980268, 2.31827224799800900953116068903, 2.41118858544775018862521200770, 3.40684771147111673326485675938, 3.61054123261623889624678982531, 4.04202540316800303734456461956, 4.27722418500781214572824406804, 5.06888163991834312506339145999, 5.26095683055226901473017944884, 5.95607206450521569803776821254, 6.21569155671181032516734739000, 6.72654241750902818200843694655, 7.16417015302239535967104674234, 7.33449156432664284571101526990, 7.74309430036937936417494119079, 8.144214142345946146944784820603, 8.227839302657972865682218318146

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