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Properties

Label 4-2790e2-1.1-c1e2-0-12
Degree $4$
Conductor $7784100$
Sign $1$
Analytic cond. $496.320$
Root an. cond. $4.71998$
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  + 2·2-s + 3·4-s − 2·5-s − 4·7-s + 4·8-s − 4·10-s − 4·11-s + 4·13-s − 8·14-s + 5·16-s − 6·20-s − 8·22-s − 4·23-s + 3·25-s + 8·26-s − 12·28-s − 16·29-s − 2·31-s + 6·32-s + 8·35-s + 4·37-s − 8·40-s − 16·43-s − 12·44-s − 8·46-s − 12·47-s − 2·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 1.41·8-s − 1.26·10-s − 1.20·11-s + 1.10·13-s − 2.13·14-s + 5/4·16-s − 1.34·20-s − 1.70·22-s − 0.834·23-s + 3/5·25-s + 1.56·26-s − 2.26·28-s − 2.97·29-s − 0.359·31-s + 1.06·32-s + 1.35·35-s + 0.657·37-s − 1.26·40-s − 2.43·43-s − 1.80·44-s − 1.17·46-s − 1.75·47-s − 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7784100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(496.320\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7784100,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_u
13$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_y
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.23.e_by
29$D_{4}$ \( 1 + 16 T + 4 p T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.29.q_em
37$D_{4}$ \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_cu
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$D_{4}$ \( 1 + 16 T + 144 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.43.q_fo
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$D_{4}$ \( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_ce
59$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_eg
61$D_{4}$ \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_fc
67$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.67.q_gs
71$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.71.a_bu
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.79.a_fe
83$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_bg
89$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_eo
97$D_{4}$ \( 1 - 8 T + 186 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_he
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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

−8.308458983552151283740693531222, −8.215513143007001949630482701972, −7.70897318495189382654863685061, −7.49403275224117371582034366162, −6.84795676934113687581015258470, −6.77202361703553807681713483901, −6.11122551504348560775803132544, −6.00470514680256687709207388431, −5.41071570330899445185925680962, −5.29397591488532094900707504043, −4.40114073446455363791218232348, −4.39821472824005136079120561370, −3.62171219375031205202388843566, −3.52215970897119357411664266983, −2.97249351614373549162830924179, −2.88748495525543559393267780226, −1.71577376471700504147179695759, −1.69515104889183192297577697036, 0, 0, 1.69515104889183192297577697036, 1.71577376471700504147179695759, 2.88748495525543559393267780226, 2.97249351614373549162830924179, 3.52215970897119357411664266983, 3.62171219375031205202388843566, 4.39821472824005136079120561370, 4.40114073446455363791218232348, 5.29397591488532094900707504043, 5.41071570330899445185925680962, 6.00470514680256687709207388431, 6.11122551504348560775803132544, 6.77202361703553807681713483901, 6.84795676934113687581015258470, 7.49403275224117371582034366162, 7.70897318495189382654863685061, 8.215513143007001949630482701972, 8.308458983552151283740693531222

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