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Properties

Label 4-2700-1.1-c1e2-0-2
Degree $4$
Conductor $2700$
Sign $1$
Analytic cond. $0.172154$
Root an. cond. $0.644138$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 8·7-s + 9-s + 12-s + 4·13-s + 16-s − 8·19-s − 8·21-s + 25-s + 27-s − 8·28-s + 16·31-s + 36-s + 4·37-s + 4·39-s − 8·43-s + 48-s + 34·49-s + 4·52-s − 8·57-s − 20·61-s − 8·63-s + 64-s − 8·67-s + 4·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.192·27-s − 1.51·28-s + 2.87·31-s + 1/6·36-s + 0.657·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s + 34/7·49-s + 0.554·52-s − 1.05·57-s − 2.56·61-s − 1.00·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.172154\)
Root analytic conductor: \(0.644138\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2700,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7472587609\)
\(L(\frac12)\) \(\approx\) \(0.7472587609\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−13.19720570989419308005927819605, −12.30986793353697911256504493943, −12.14360527021257208948458691105, −10.91913531404329383721387224500, −10.39160509435207624542212960871, −9.880541566397139490214892173475, −9.305587122869086271308905422277, −8.644229019078172602684577487278, −7.941231322257043910223245152113, −6.73106470746564203527567325446, −6.42217617652666865799421983967, −6.11397905422548756742635703073, −4.32687879913691587957737883725, −3.36585804145949552210873743484, −2.74628747875934588873509462774, 2.74628747875934588873509462774, 3.36585804145949552210873743484, 4.32687879913691587957737883725, 6.11397905422548756742635703073, 6.42217617652666865799421983967, 6.73106470746564203527567325446, 7.941231322257043910223245152113, 8.644229019078172602684577487278, 9.305587122869086271308905422277, 9.880541566397139490214892173475, 10.39160509435207624542212960871, 10.91913531404329383721387224500, 12.14360527021257208948458691105, 12.30986793353697911256504493943, 13.19720570989419308005927819605

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