Properties

Label 4-63e2-1.1-c1e2-0-3
Degree $4$
Conductor $3969$
Sign $1$
Analytic cond. $0.253066$
Root an. cond. $0.709265$
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  + 2·4-s − 7-s − 14·13-s + 7·19-s + 5·25-s − 2·28-s + 7·31-s + 37-s + 10·43-s − 6·49-s − 28·52-s − 14·61-s − 8·64-s − 11·67-s + 7·73-s + 14·76-s + 13·79-s + 14·91-s + 28·97-s + 10·100-s + 7·103-s − 17·109-s + 11·121-s + 14·124-s + ⋯
L(s)  = 1  + 4-s − 0.377·7-s − 3.88·13-s + 1.60·19-s + 25-s − 0.377·28-s + 1.25·31-s + 0.164·37-s + 1.52·43-s − 6/7·49-s − 3.88·52-s − 1.79·61-s − 64-s − 1.34·67-s + 0.819·73-s + 1.60·76-s + 1.46·79-s + 1.46·91-s + 2.84·97-s + 100-s + 0.689·103-s − 1.62·109-s + 121-s + 1.25·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.253066\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3969,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8528974404\)
\(L(\frac12)\) \(\approx\) \(0.8528974404\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.13.o_cx
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.53.a_acb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.l_cc
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ah_ay
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−15.10391512386323464488553043475, −14.89204274700472610334583207136, −14.19759421423389586027241204546, −13.81857004699226797923647639741, −12.75045934194220757550721874333, −12.43916690255645319183688183218, −11.76131590485914725834823505798, −11.73034383769830250715065081985, −10.57513541047567563154004230669, −10.20281678875964561378998763774, −9.438431329900591574938370924126, −9.250577143948027650400233365546, −7.70102303608271730101848054391, −7.56481020328747985759872945590, −6.95697683731745953258666267602, −6.22998923426805380952819476845, −5.06428004548084692695021768767, −4.71954527606832227016107159929, −2.95585781887631646524441704782, −2.45311402063529036694048344147, 2.45311402063529036694048344147, 2.95585781887631646524441704782, 4.71954527606832227016107159929, 5.06428004548084692695021768767, 6.22998923426805380952819476845, 6.95697683731745953258666267602, 7.56481020328747985759872945590, 7.70102303608271730101848054391, 9.250577143948027650400233365546, 9.438431329900591574938370924126, 10.20281678875964561378998763774, 10.57513541047567563154004230669, 11.73034383769830250715065081985, 11.76131590485914725834823505798, 12.43916690255645319183688183218, 12.75045934194220757550721874333, 13.81857004699226797923647639741, 14.19759421423389586027241204546, 14.89204274700472610334583207136, 15.10391512386323464488553043475

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