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Properties

Label 4-5200e2-1.1-c1e2-0-12
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  + 3-s − 7-s − 9-s + 2·11-s − 2·13-s + 17-s − 2·19-s − 21-s − 16·23-s − 4·29-s − 8·31-s + 2·33-s − 7·37-s − 2·39-s + 2·41-s + 15·43-s − 13·47-s − 9·49-s + 51-s + 2·53-s − 2·57-s − 2·59-s + 14·61-s + 63-s − 2·67-s − 16·69-s + 3·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.242·17-s − 0.458·19-s − 0.218·21-s − 3.33·23-s − 0.742·29-s − 1.43·31-s + 0.348·33-s − 1.15·37-s − 0.320·39-s + 0.312·41-s + 2.28·43-s − 1.89·47-s − 9/7·49-s + 0.140·51-s + 0.274·53-s − 0.264·57-s − 0.260·59-s + 1.79·61-s + 0.125·63-s − 0.244·67-s − 1.92·69-s + 0.356·71-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$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
7$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_k
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_g
17$D_{4}$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_ae
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_w
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$D_{4}$ \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_bw
41$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_co
43$D_{4}$ \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.43.ap_fi
47$D_{4}$ \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.47.n_du
53$D_{4}$ \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_dm
59$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_dy
61$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.61.ao_fy
67$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_eo
71$D_{4}$ \( 1 - 3 T + 106 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.71.ad_ec
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_fe
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.97.a_ew
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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

−8.026642507422669761059204455682, −7.86061233227576706915467278529, −7.25546384018861871296420345278, −7.07707774038457988039724322452, −6.59862760049216744189215010458, −6.16142424630787020053745722568, −5.92981294406266044089713846339, −5.54990092800269728184128789117, −5.22676006319721308160982734219, −4.54406800678627966732615865159, −4.27107884601705859727438774753, −3.83792058552520780745080226009, −3.45526606635599252297821746820, −3.28519759032898692152690264158, −2.34675726386360868728801056231, −2.28125565019758375297421050789, −1.82242405707376003780275513056, −1.18694149657762043370744291289, 0, 0, 1.18694149657762043370744291289, 1.82242405707376003780275513056, 2.28125565019758375297421050789, 2.34675726386360868728801056231, 3.28519759032898692152690264158, 3.45526606635599252297821746820, 3.83792058552520780745080226009, 4.27107884601705859727438774753, 4.54406800678627966732615865159, 5.22676006319721308160982734219, 5.54990092800269728184128789117, 5.92981294406266044089713846339, 6.16142424630787020053745722568, 6.59862760049216744189215010458, 7.07707774038457988039724322452, 7.25546384018861871296420345278, 7.86061233227576706915467278529, 8.026642507422669761059204455682

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