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Properties

Label 4-1440e2-1.1-c1e2-0-13
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
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  − 3·3-s + 5-s − 5·7-s + 6·9-s + 2·11-s + 6·13-s − 3·15-s + 16·17-s + 8·19-s + 15·21-s − 3·23-s − 9·27-s − 7·29-s + 4·31-s − 6·33-s − 5·35-s − 8·37-s − 18·39-s − 5·41-s − 8·43-s + 6·45-s + 47-s + 7·49-s − 48·51-s + 4·53-s + 2·55-s − 24·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s − 1.88·7-s + 2·9-s + 0.603·11-s + 1.66·13-s − 0.774·15-s + 3.88·17-s + 1.83·19-s + 3.27·21-s − 0.625·23-s − 1.73·27-s − 1.29·29-s + 0.718·31-s − 1.04·33-s − 0.845·35-s − 1.31·37-s − 2.88·39-s − 0.780·41-s − 1.21·43-s + 0.894·45-s + 0.145·47-s + 49-s − 6.72·51-s + 0.549·53-s + 0.269·55-s − 3.17·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260667588\)
\(L(\frac12)\) \(\approx\) \(1.260667588\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.f_s
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.17.aq_du
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2^2$ \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.29.h_u
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_aq
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2^2$ \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) 2.47.ab_abu
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_acg
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.73.q_ic
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_v
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) 2.83.b_ade
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−9.788675505065638279740169601691, −9.646434563821847536543816398256, −9.180383913935014064714402943114, −8.569101179506431051513367208375, −7.976565115662929613618817037903, −7.63188340315851142799106847976, −7.09828196050498414289953152241, −6.77929373783523351897564452638, −6.34092126675569775696386139256, −5.86896892793052493613509974411, −5.77722611040151075121486977705, −5.29551525967771654862793329106, −5.14198440652983944015443441375, −4.01620889716993095072539899835, −3.67346766278826708870417559443, −3.27060311915371015399232914345, −3.08091811530681106097017203688, −1.48088463375938448566969119497, −1.35816500672471524511137407558, −0.58699944297423921081919902797, 0.58699944297423921081919902797, 1.35816500672471524511137407558, 1.48088463375938448566969119497, 3.08091811530681106097017203688, 3.27060311915371015399232914345, 3.67346766278826708870417559443, 4.01620889716993095072539899835, 5.14198440652983944015443441375, 5.29551525967771654862793329106, 5.77722611040151075121486977705, 5.86896892793052493613509974411, 6.34092126675569775696386139256, 6.77929373783523351897564452638, 7.09828196050498414289953152241, 7.63188340315851142799106847976, 7.976565115662929613618817037903, 8.569101179506431051513367208375, 9.180383913935014064714402943114, 9.646434563821847536543816398256, 9.788675505065638279740169601691

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