Properties

Label 4-1536e2-1.1-c1e2-0-23
Degree $4$
Conductor $2359296$
Sign $1$
Analytic cond. $150.430$
Root an. cond. $3.50214$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 4·7-s + 3·9-s + 4·11-s − 8·15-s − 4·17-s − 8·21-s − 8·23-s + 4·25-s + 4·27-s − 4·29-s − 12·31-s + 8·33-s + 16·35-s + 8·37-s − 12·41-s − 8·43-s − 12·45-s − 8·47-s − 8·51-s − 4·53-s − 16·55-s − 8·59-s + 8·61-s − 12·63-s − 16·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 1.51·7-s + 9-s + 1.20·11-s − 2.06·15-s − 0.970·17-s − 1.74·21-s − 1.66·23-s + 4/5·25-s + 0.769·27-s − 0.742·29-s − 2.15·31-s + 1.39·33-s + 2.70·35-s + 1.31·37-s − 1.87·41-s − 1.21·43-s − 1.78·45-s − 1.16·47-s − 1.12·51-s − 0.549·53-s − 2.15·55-s − 1.04·59-s + 1.02·61-s − 1.51·63-s − 1.95·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2359296\)    =    \(2^{18} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(150.430\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2359296,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_m
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_q
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_g
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_cc
29$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_ci
31$D_{4}$ \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_ds
37$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_cg
41$C_4$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_di
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cs
47$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_bm
53$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_m
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$D_{4}$ \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_ec
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$D_{4}$ \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.71.y_ks
73$D_{4}$ \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fa
79$D_{4}$ \( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.79.u_jg
83$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_bq
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_cs
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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

−9.199380344620141752829390738797, −8.843756031525953434722316280053, −8.372224999459781424147248768925, −8.235360946977928144348023861178, −7.51599449209749458665495597989, −7.44259255841570297167411143135, −6.92439011991927621076312483037, −6.61740660060842387249899503635, −5.99362123127882086020716226922, −5.84456648570674083849255551281, −4.69727015041853221734010932647, −4.49062298262009125730221335898, −3.87420217487797354532728258288, −3.73729633873441391363326595861, −3.19246285305603941054627274427, −3.05904623629176782130902699142, −1.85447277007451169258533306237, −1.72888012606182572417067796976, 0, 0, 1.72888012606182572417067796976, 1.85447277007451169258533306237, 3.05904623629176782130902699142, 3.19246285305603941054627274427, 3.73729633873441391363326595861, 3.87420217487797354532728258288, 4.49062298262009125730221335898, 4.69727015041853221734010932647, 5.84456648570674083849255551281, 5.99362123127882086020716226922, 6.61740660060842387249899503635, 6.92439011991927621076312483037, 7.44259255841570297167411143135, 7.51599449209749458665495597989, 8.235360946977928144348023861178, 8.372224999459781424147248768925, 8.843756031525953434722316280053, 9.199380344620141752829390738797

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