↑

Properties

Label 4-12e6-1.1-c1e2-0-28
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $190.388$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 12·11-s − 6·17-s − 4·19-s − 3·25-s + 27-s − 12·33-s + 6·41-s − 10·43-s − 5·49-s − 6·51-s − 4·57-s − 12·59-s − 15·67-s + 3·73-s − 3·75-s + 81-s + 6·83-s − 2·97-s − 12·99-s − 24·107-s − 12·113-s + 86·121-s + 6·123-s + 127-s − 10·129-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 3.61·11-s − 1.45·17-s − 0.917·19-s − 3/5·25-s + 0.192·27-s − 2.08·33-s + 0.937·41-s − 1.52·43-s − 5/7·49-s − 0.840·51-s − 0.529·57-s − 1.56·59-s − 1.83·67-s + 0.351·73-s − 0.346·75-s + 1/9·81-s + 0.658·83-s − 0.203·97-s − 1.20·99-s − 2.32·107-s − 1.12·113-s + 7.81·121-s + 0.541·123-s + 0.0887·127-s − 0.880·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(190.388\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2985984,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.g_bq
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.e_bh
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.23.a_ad
29$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.29.a_abp
31$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.31.a_bp
37$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.37.a_at
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ag_co
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.k_cx
47$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.47.a_ax
53$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.53.a_br
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.m_fu
61$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.61.a_ax
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.p_go
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.73.ad_ey
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.ag_cc
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.c_gd
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

−7.36321726317807501184616380221, −6.70268789390920637374853175540, −6.37970033351989965088816874826, −5.85612955700901692580362736561, −5.32089518185112969883117040897, −5.05259456083182728505696851730, −4.55532820773605565791763762659, −4.25517173980391237504918577673, −3.47137407265729972950729761473, −2.93726929504042665544471613466, −2.51920889196178102493772707276, −2.26285676637873998336581522478, −1.57920733252426140993859887263, 0, 0, 1.57920733252426140993859887263, 2.26285676637873998336581522478, 2.51920889196178102493772707276, 2.93726929504042665544471613466, 3.47137407265729972950729761473, 4.25517173980391237504918577673, 4.55532820773605565791763762659, 5.05259456083182728505696851730, 5.32089518185112969883117040897, 5.85612955700901692580362736561, 6.37970033351989965088816874826, 6.70268789390920637374853175540, 7.36321726317807501184616380221

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