↑

Properties

Label 4-24e4-1.1-c1e2-0-32
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s + 8·13-s + 12·23-s + 2·25-s + 4·37-s + 4·47-s + 10·49-s − 16·59-s − 4·61-s − 4·71-s − 4·73-s − 28·83-s + 4·97-s − 8·107-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯
L(s)  = 1  + 1.20·11-s + 2.21·13-s + 2.50·23-s + 2/5·25-s + 0.657·37-s + 0.583·47-s + 10/7·49-s − 2.08·59-s − 0.512·61-s − 0.474·71-s − 0.468·73-s − 3.07·83-s + 0.406·97-s − 0.773·107-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-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: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.487247569\)
\(L(\frac12)\) \(\approx\) \(2.487247569\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
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$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ai_bm
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.am_da
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.31.a_cc
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.41.a_bq
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.43.a_abq
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.ae_ck
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.q_gk
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.e_eg
67$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.67.a_acg
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.e_fm
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.e_fe
79$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.79.a_ak
83$C_2$$\times$$C_2$ \( ( 1 + 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.bc_nu
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.89.a_fa
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ae_cc
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

−8.849992037896896618448988127874, −8.516429707022264165941143619278, −7.85915389556093308374638347098, −7.28041937899198388112679911274, −6.84336689814190172896956344878, −6.48589505307606971197543984812, −5.83364779378410718614120511305, −5.67615307631966384647483489302, −4.73826545171632262024239382998, −4.34790186495541683284184782024, −3.73593660003712570123542581756, −3.23050813707198963747749368953, −2.65787015489133012574937680489, −1.35142392900134993828670961759, −1.15646370652108519773326923121, 1.15646370652108519773326923121, 1.35142392900134993828670961759, 2.65787015489133012574937680489, 3.23050813707198963747749368953, 3.73593660003712570123542581756, 4.34790186495541683284184782024, 4.73826545171632262024239382998, 5.67615307631966384647483489302, 5.83364779378410718614120511305, 6.48589505307606971197543984812, 6.84336689814190172896956344878, 7.28041937899198388112679911274, 7.85915389556093308374638347098, 8.516429707022264165941143619278, 8.849992037896896618448988127874

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