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Properties

Label 4-894348-1.1-c1e2-0-59
Degree $4$
Conductor $894348$
Sign $-1$
Analytic cond. $57.0244$
Root an. cond. $2.74799$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 2·7-s + 9-s + 12-s − 2·13-s + 16-s − 8·19-s + 2·21-s − 6·25-s + 27-s + 2·28-s − 16·31-s + 36-s + 20·37-s − 2·39-s + 8·43-s + 48-s + 3·49-s − 2·52-s − 8·57-s − 12·61-s + 2·63-s + 64-s − 16·67-s − 20·73-s − 6·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 1.83·19-s + 0.436·21-s − 6/5·25-s + 0.192·27-s + 0.377·28-s − 2.87·31-s + 1/6·36-s + 3.28·37-s − 0.320·39-s + 1.21·43-s + 0.144·48-s + 3/7·49-s − 0.277·52-s − 1.05·57-s − 1.53·61-s + 0.251·63-s + 1/8·64-s − 1.95·67-s − 2.34·73-s − 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(894348\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(57.0244\)
Root analytic conductor: \(2.74799\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 894348,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
7$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−7.927583890916278548349975461285, −7.53512261450424986470295233692, −7.31420502617006761020558985636, −6.70956799882972482046383355029, −6.02469114263822030525417968426, −5.87416482309586917103143441505, −5.34066639773140823665957604454, −4.51751350141054354627536433428, −4.08389192736738734823347258017, −4.05735746111134485794177861946, −2.81336235055681255128786870399, −2.70225474965229643722185344974, −1.85528951457630629269007018576, −1.52032901507690394890240915480, 0, 1.52032901507690394890240915480, 1.85528951457630629269007018576, 2.70225474965229643722185344974, 2.81336235055681255128786870399, 4.05735746111134485794177861946, 4.08389192736738734823347258017, 4.51751350141054354627536433428, 5.34066639773140823665957604454, 5.87416482309586917103143441505, 6.02469114263822030525417968426, 6.70956799882972482046383355029, 7.31420502617006761020558985636, 7.53512261450424986470295233692, 7.927583890916278548349975461285

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