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Properties

Label 4-5202e2-1.1-c1e2-0-12
Degree $4$
Conductor $27060804$
Sign $1$
Analytic cond. $1725.42$
Root an. cond. $6.44501$
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 − 4·8-s − 12·13-s + 5·16-s + 8·19-s − 2·25-s + 24·26-s − 6·32-s − 16·38-s + 24·43-s − 20·47-s − 12·49-s + 4·50-s − 36·52-s + 12·53-s + 7·64-s + 8·67-s + 24·76-s − 32·83-s − 48·86-s − 24·89-s + 40·94-s + 24·98-s − 6·100-s − 28·101-s − 4·103-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 3.32·13-s + 5/4·16-s + 1.83·19-s − 2/5·25-s + 4.70·26-s − 1.06·32-s − 2.59·38-s + 3.65·43-s − 2.91·47-s − 1.71·49-s + 0.565·50-s − 4.99·52-s + 1.64·53-s + 7/8·64-s + 0.977·67-s + 2.75·76-s − 3.51·83-s − 5.17·86-s − 2.54·89-s + 4.12·94-s + 2.42·98-s − 3/5·100-s − 2.78·101-s − 0.394·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27060804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27060804\)    =    \(2^{2} \cdot 3^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1725.42\)
Root analytic conductor: \(6.44501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27060804,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
17 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.23.a_bc
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.29.a_by
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.41.a_dc
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.47.u_hm
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.61.a_ek
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \) 2.71.a_fk
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \) 2.73.a_fo
79$C_2^2$ \( 1 + 108 T^{2} + p^{2} T^{4} \) 2.79.a_ee
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.83.bg_qg
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \) 2.97.a_abw
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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

−7.975651649059871150276668106585, −7.76121043440256224935141202204, −7.34039359006153008722328180034, −7.13970990464356669522366505502, −6.82490242113671407619783578058, −6.53663761817595037057476578508, −5.73256689783162256392863487541, −5.57424753588337693324526660572, −5.21395284705621846276817132426, −4.87268524717272603417978228361, −4.22606347424818622690974268455, −4.04640825984316060567006853798, −3.01205791175435709148495464794, −3.00844145654009425011917256485, −2.49839887865641368095107721357, −2.19093396909125320103006053180, −1.44873419868143909399937424816, −1.06029608974008040071795001976, 0, 0, 1.06029608974008040071795001976, 1.44873419868143909399937424816, 2.19093396909125320103006053180, 2.49839887865641368095107721357, 3.00844145654009425011917256485, 3.01205791175435709148495464794, 4.04640825984316060567006853798, 4.22606347424818622690974268455, 4.87268524717272603417978228361, 5.21395284705621846276817132426, 5.57424753588337693324526660572, 5.73256689783162256392863487541, 6.53663761817595037057476578508, 6.82490242113671407619783578058, 7.13970990464356669522366505502, 7.34039359006153008722328180034, 7.76121043440256224935141202204, 7.975651649059871150276668106585

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