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Properties

Label 4-24e4-1.1-c1e2-0-35
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
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  + 3·5-s + 7-s − 3·9-s + 3·11-s − 13-s + 12·17-s + 8·19-s + 3·23-s + 5·25-s + 3·29-s − 5·31-s + 3·35-s − 4·37-s − 3·41-s − 43-s − 9·45-s + 9·47-s + 7·49-s + 12·53-s + 9·55-s − 3·59-s − 13·61-s − 3·63-s − 3·65-s − 7·67-s − 24·71-s − 20·73-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 9-s + 0.904·11-s − 0.277·13-s + 2.91·17-s + 1.83·19-s + 0.625·23-s + 25-s + 0.557·29-s − 0.898·31-s + 0.507·35-s − 0.657·37-s − 0.468·41-s − 0.152·43-s − 1.34·45-s + 1.31·47-s + 49-s + 1.64·53-s + 1.21·55-s − 0.390·59-s − 1.66·61-s − 0.377·63-s − 0.372·65-s − 0.855·67-s − 2.84·71-s − 2.34·73-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.904380311\)
\(L(\frac12)\) \(\approx\) \(2.904380311\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_am
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_au
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_ag
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.n_ee
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_as
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.79.l_bq
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.97.l_y
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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

−10.66373857453799648276687311273, −10.55464941196143113002808641854, −9.882443508847157671468397338947, −9.820237493180040822404382631213, −9.164206727321139778469551052702, −8.799860906056615216621763438136, −8.558602238031447478656016439292, −7.62746150735450034914185310860, −7.33085822679948347735005622646, −7.17224822521579406196943341796, −6.05664735282021213241486802974, −5.86423560611527013229170910278, −5.53086686710374051876759273318, −5.16798750198110613568592580835, −4.45125123198266736868986155425, −3.53068864789014727665592923665, −3.10315719359876178857067154391, −2.65666174448507744543953211222, −1.45898962843297050057826204341, −1.20494341501628432479218500582, 1.20494341501628432479218500582, 1.45898962843297050057826204341, 2.65666174448507744543953211222, 3.10315719359876178857067154391, 3.53068864789014727665592923665, 4.45125123198266736868986155425, 5.16798750198110613568592580835, 5.53086686710374051876759273318, 5.86423560611527013229170910278, 6.05664735282021213241486802974, 7.17224822521579406196943341796, 7.33085822679948347735005622646, 7.62746150735450034914185310860, 8.558602238031447478656016439292, 8.799860906056615216621763438136, 9.164206727321139778469551052702, 9.820237493180040822404382631213, 9.882443508847157671468397338947, 10.55464941196143113002808641854, 10.66373857453799648276687311273

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