Properties

Label 4-591872-1.1-c1e2-0-14
Degree $4$
Conductor $591872$
Sign $1$
Analytic cond. $37.7382$
Root an. cond. $2.47853$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 8·11-s − 2·17-s − 8·19-s − 10·25-s − 4·41-s + 8·43-s − 10·49-s − 8·59-s − 8·67-s − 12·73-s + 27·81-s + 24·83-s − 20·89-s − 20·97-s + 48·99-s + 8·107-s + 20·113-s + 26·121-s + ⋯
L(s)  = 1  − 2·9-s − 2.41·11-s − 0.485·17-s − 1.83·19-s − 2·25-s − 0.624·41-s + 1.21·43-s − 1.42·49-s − 1.04·59-s − 0.977·67-s − 1.40·73-s + 3·81-s + 2.63·83-s − 2.11·89-s − 2.03·97-s + 4.82·99-s + 0.773·107-s + 1.88·113-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 591872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 591872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(591872\)    =    \(2^{11} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(37.7382\)
Root analytic conductor: \(2.47853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 591872,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−7.949465060430367136751257620846, −7.81365907337972927586940197328, −7.14094121934417456923588053574, −6.45486383530876632990023364782, −5.92602088890819392149162970339, −5.79350053985432556666939907600, −5.27206004086745820789620778593, −4.68811629121503471986666453564, −4.25100997583610576744300309881, −3.37777254090361242162912900058, −2.92399094991670175082934692290, −2.34190413882648615913196631793, −2.00061117095674896363863230854, 0, 0, 2.00061117095674896363863230854, 2.34190413882648615913196631793, 2.92399094991670175082934692290, 3.37777254090361242162912900058, 4.25100997583610576744300309881, 4.68811629121503471986666453564, 5.27206004086745820789620778593, 5.79350053985432556666939907600, 5.92602088890819392149162970339, 6.45486383530876632990023364782, 7.14094121934417456923588053574, 7.81365907337972927586940197328, 7.949465060430367136751257620846

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