↑

Properties

Label 4-24e4-1.1-c1e2-0-48
Degree $4$
Conductor $331776$
Sign $-1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s − 4·13-s − 12·23-s − 4·25-s + 4·37-s + 4·47-s − 2·49-s + 8·59-s − 4·61-s − 4·71-s − 16·73-s − 4·83-s − 8·97-s + 16·107-s − 12·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s − 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯
L(s)  = 1  + 1.20·11-s − 1.10·13-s − 2.50·23-s − 4/5·25-s + 0.657·37-s + 0.583·47-s − 2/7·49-s + 1.04·59-s − 0.512·61-s − 0.474·71-s − 1.87·73-s − 0.439·83-s − 0.812·97-s + 1.54·107-s − 1.14·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.33·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ae_w
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.e_ba
17$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.17.a_ai
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.m_da
29$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \) 2.29.a_aca
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.31.a_s
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.41.a_ay
43$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.43.a_bq
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.ae_ck
53$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.53.a_ci
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ai_eo
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.e_eg
67$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.67.a_cw
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.e_bu
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.73.q_hm
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.79.a_by
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.e_aba
89$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.89.a_ei
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.i_co
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

−8.508890927228317107809155804325, −8.053531283266921403399232216356, −7.58362468779951906752252314034, −7.23581144856406587183158995420, −6.61804890147561498199371851340, −6.10217232780957668770129648496, −5.83252765072089165649169088141, −5.17530886323472951818277662326, −4.45114320120815774261463479937, −4.08546297328893572177165756463, −3.65956894261319686677630972148, −2.73025569532436935523506079390, −2.14349042559593556748670331493, −1.41065699043739306062040335688, 0, 1.41065699043739306062040335688, 2.14349042559593556748670331493, 2.73025569532436935523506079390, 3.65956894261319686677630972148, 4.08546297328893572177165756463, 4.45114320120815774261463479937, 5.17530886323472951818277662326, 5.83252765072089165649169088141, 6.10217232780957668770129648496, 6.61804890147561498199371851340, 7.23581144856406587183158995420, 7.58362468779951906752252314034, 8.053531283266921403399232216356, 8.508890927228317107809155804325

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