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Properties

Label 4-5200e2-1.1-c1e2-0-8
Degree $4$
Conductor $27040000$
Sign $1$
Analytic cond. $1724.09$
Root an. cond. $6.44377$
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 − 2·11-s − 2·13-s − 8·17-s + 2·19-s + 2·23-s + 2·27-s + 12·29-s − 10·31-s + 4·33-s + 4·39-s + 8·41-s − 18·43-s + 16·47-s − 2·49-s + 16·51-s + 8·53-s − 4·57-s + 6·59-s − 4·61-s − 12·67-s − 4·69-s − 22·71-s − 8·73-s + 20·79-s − 81-s − 24·87-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.603·11-s − 0.554·13-s − 1.94·17-s + 0.458·19-s + 0.417·23-s + 0.384·27-s + 2.22·29-s − 1.79·31-s + 0.696·33-s + 0.640·39-s + 1.24·41-s − 2.74·43-s + 2.33·47-s − 2/7·49-s + 2.24·51-s + 1.09·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s − 1.46·67-s − 0.481·69-s − 2.61·71-s − 0.936·73-s + 2.25·79-s − 1/9·81-s − 2.57·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27040000\)    =    \(2^{8} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1724.09\)
Root analytic conductor: \(6.44377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27040000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_u
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bm
19$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_bk
23$D_{4}$ \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_bs
29$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_de
31$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_ci
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_di
43$D_{4}$ \( 1 + 18 T + 164 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.43.s_gi
47$D_{4}$ \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.47.aq_fq
53$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_eg
59$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ca
61$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_s
67$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_es
71$D_{4}$ \( 1 + 22 T + 260 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.71.w_ka
73$D_{4}$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_ek
79$D_{4}$ \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.79.au_jm
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gk
97$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_g
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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

−7.964440243132471251528253197458, −7.48533106312880452912422930582, −7.32579410472412642309612913204, −6.91833388743242358929560553309, −6.42162316749498655354221516106, −6.39902656240681212225221274967, −5.69362044319817163720997871446, −5.57468121123883864828390769595, −5.16537759416049391161042639985, −4.80103717676258842866044129916, −4.35974320464378628524015949108, −4.20725077451448409958673450553, −3.45467662026697687670093474337, −3.02950296503356317686293103477, −2.47028988187206036669265766765, −2.32749012593198099361295159859, −1.52956659879690338592356178889, −0.948036892060061659836318088949, 0, 0, 0.948036892060061659836318088949, 1.52956659879690338592356178889, 2.32749012593198099361295159859, 2.47028988187206036669265766765, 3.02950296503356317686293103477, 3.45467662026697687670093474337, 4.20725077451448409958673450553, 4.35974320464378628524015949108, 4.80103717676258842866044129916, 5.16537759416049391161042639985, 5.57468121123883864828390769595, 5.69362044319817163720997871446, 6.39902656240681212225221274967, 6.42162316749498655354221516106, 6.91833388743242358929560553309, 7.32579410472412642309612913204, 7.48533106312880452912422930582, 7.964440243132471251528253197458

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