↑

Properties

Label 4-394272-1.1-c1e2-0-11
Degree $4$
Conductor $394272$
Sign $-1$
Analytic cond. $25.1391$
Root an. cond. $2.23917$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 9-s + 2·13-s + 16-s − 6·17-s + 18-s − 10·25-s + 2·26-s − 8·29-s + 32-s − 6·34-s + 36-s − 2·37-s − 20·41-s − 5·49-s − 10·50-s + 2·52-s − 2·53-s − 8·58-s + 4·61-s + 64-s − 6·68-s + 72-s − 6·73-s − 2·74-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 2·25-s + 0.392·26-s − 1.48·29-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 3.12·41-s − 5/7·49-s − 1.41·50-s + 0.277·52-s − 0.274·53-s − 1.05·58-s + 0.512·61-s + 1/8·64-s − 0.727·68-s + 0.117·72-s − 0.702·73-s − 0.232·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 394272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(394272\)    =    \(2^{5} \cdot 3^{2} \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(25.1391\)
Root analytic conductor: \(2.23917\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 394272,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.a_v
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.a_bd
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.53.c_ed
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.a_fa
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.73.g_fz
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.325592580285100754277675037249, −8.093256336428860279579050698932, −7.23351767004553543452225501388, −7.07433016066728121125499972487, −6.58594129796607005303022052317, −5.85671815365648243578194470848, −5.79702284200659259065474780337, −4.93990077281798148757107408401, −4.62546171656297684808475214190, −3.81933209527513554138512512650, −3.69745098069647884007817791582, −2.91055328661416272205710045681, −1.84137075722148284400280530836, −1.81998707337995438672490638604, 0, 1.81998707337995438672490638604, 1.84137075722148284400280530836, 2.91055328661416272205710045681, 3.69745098069647884007817791582, 3.81933209527513554138512512650, 4.62546171656297684808475214190, 4.93990077281798148757107408401, 5.79702284200659259065474780337, 5.85671815365648243578194470848, 6.58594129796607005303022052317, 7.07433016066728121125499972487, 7.23351767004553543452225501388, 8.093256336428860279579050698932, 8.325592580285100754277675037249

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