Properties

Label 4-1536e2-1.1-c1e2-0-21
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.ae_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.e_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.ae_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.i_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.ai_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.ae_m
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$D_{4}$ \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_ec
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_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.ae_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

−9.260432412887894749807029307421, −9.227805000838494247216220735257, −8.387363131534810749196066443307, −8.249573302753600809777390498565, −7.28662420213233525916928170900, −7.23135634455767805074048719988, −6.57293713105416332314713311281, −6.37934432897306394673822402913, −5.92765373912236172303736962828, −5.69593168188394116934774112579, −5.21095925852256138048589516863, −4.98799072583447757499765692311, −4.18712588052605308209395746625, −3.71787849065387894947905118136, −3.12536603370223893837381633932, −2.47282770308882758013700274778, −1.95730900205362894878313377014, −1.56426393760888427293250305267, 0, 0, 1.56426393760888427293250305267, 1.95730900205362894878313377014, 2.47282770308882758013700274778, 3.12536603370223893837381633932, 3.71787849065387894947905118136, 4.18712588052605308209395746625, 4.98799072583447757499765692311, 5.21095925852256138048589516863, 5.69593168188394116934774112579, 5.92765373912236172303736962828, 6.37934432897306394673822402913, 6.57293713105416332314713311281, 7.23135634455767805074048719988, 7.28662420213233525916928170900, 8.249573302753600809777390498565, 8.387363131534810749196066443307, 9.227805000838494247216220735257, 9.260432412887894749807029307421

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