Properties

Label 4-581184-1.1-c1e2-0-1
Degree $4$
Conductor $581184$
Sign $-1$
Analytic cond. $37.0567$
Root an. cond. $2.46727$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + 13-s + 19-s + 25-s + 4·27-s − 4·31-s − 7·37-s − 2·39-s + 17·43-s − 10·49-s − 2·57-s − 7·61-s − 12·67-s − 7·73-s − 2·75-s − 7·79-s − 11·81-s + 8·93-s + 15·97-s − 16·103-s + 22·109-s + 14·111-s + 117-s + 10·121-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 0.277·13-s + 0.229·19-s + 1/5·25-s + 0.769·27-s − 0.718·31-s − 1.15·37-s − 0.320·39-s + 2.59·43-s − 1.42·49-s − 0.264·57-s − 0.896·61-s − 1.46·67-s − 0.819·73-s − 0.230·75-s − 0.787·79-s − 1.22·81-s + 0.829·93-s + 1.52·97-s − 1.57·103-s + 2.10·109-s + 1.32·111-s + 0.0924·117-s + 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 581184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 581184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(581184\)    =    \(2^{6} \cdot 3^{2} \cdot 1009\)
Sign: $-1$
Analytic conductor: \(37.0567\)
Root analytic conductor: \(2.46727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 581184,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
1009$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 46 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) 2.13.ab_ba
17$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.17.a_ae
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ab_bk
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \) 2.29.a_abv
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.e_ck
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.h_ce
41$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \) 2.41.a_abr
43$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.43.ar_fw
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.47.a_de
53$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.53.a_bg
59$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.59.a_abo
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.h_bs
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.67.m_gf
71$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.71.a_cw
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.73.h_ga
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.h_fu
83$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.83.a_eg
89$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.89.a_dw
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.ap_gw
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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

−8.202741108228490129597075796040, −7.58114537532604372124805893615, −7.29509030229603541832274632713, −6.81557315012418195084836739173, −6.20966919235154791869497646247, −5.89741473052595736606368755126, −5.57742035985541333610868371287, −4.90063822383271781452146519360, −4.60068441561639828402739528966, −3.95854169351807564784346778355, −3.29395055167982143092190391452, −2.75231051843497608569693497474, −1.84658378022775822430616002878, −1.06852466750769776448809655923, 0, 1.06852466750769776448809655923, 1.84658378022775822430616002878, 2.75231051843497608569693497474, 3.29395055167982143092190391452, 3.95854169351807564784346778355, 4.60068441561639828402739528966, 4.90063822383271781452146519360, 5.57742035985541333610868371287, 5.89741473052595736606368755126, 6.20966919235154791869497646247, 6.81557315012418195084836739173, 7.29509030229603541832274632713, 7.58114537532604372124805893615, 8.202741108228490129597075796040

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