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Properties

Label 4-453152-1.1-c1e2-0-8
Degree $4$
Conductor $453152$
Sign $1$
Analytic cond. $28.8933$
Root an. cond. $2.31845$
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-s + 4-s − 4·5-s − 8-s − 6·9-s + 4·10-s + 16-s − 2·17-s + 6·18-s − 4·20-s + 2·25-s − 32-s + 2·34-s − 6·36-s − 8·37-s + 4·40-s − 12·41-s + 24·45-s + 49-s − 2·50-s − 12·53-s + 20·61-s + 64-s − 2·68-s + 6·72-s − 20·73-s + 8·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s − 2·9-s + 1.26·10-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.894·20-s + 2/5·25-s − 0.176·32-s + 0.342·34-s − 36-s − 1.31·37-s + 0.632·40-s − 1.87·41-s + 3.57·45-s + 1/7·49-s − 0.282·50-s − 1.64·53-s + 2.56·61-s + 1/8·64-s − 0.242·68-s + 0.707·72-s − 2.34·73-s + 0.929·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(453152\)    =    \(2^{5} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(28.8933\)
Root analytic conductor: \(2.31845\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 453152,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( 1 + T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−8.080485225562098313286905351335, −8.024463406411413208815234536452, −7.30473635612141536821000906080, −6.76626773482443698632812811275, −6.55401262849968600889826370986, −5.62672058490049396710575997042, −5.47236332424513932239582971582, −4.77922727113164880334613207685, −3.96511449610299471402331111472, −3.68463820290009014802243979664, −3.01385706794688684014407035010, −2.56493236008968775596152231873, −1.56306199757045232120398581261, 0, 0, 1.56306199757045232120398581261, 2.56493236008968775596152231873, 3.01385706794688684014407035010, 3.68463820290009014802243979664, 3.96511449610299471402331111472, 4.77922727113164880334613207685, 5.47236332424513932239582971582, 5.62672058490049396710575997042, 6.55401262849968600889826370986, 6.76626773482443698632812811275, 7.30473635612141536821000906080, 8.024463406411413208815234536452, 8.080485225562098313286905351335

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