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Properties

Label 4-2160e2-1.1-c1e2-0-11
Degree $4$
Conductor $4665600$
Sign $1$
Analytic cond. $297.482$
Root an. cond. $4.15303$
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  − 2·5-s + 10·11-s − 12·19-s − 25-s + 18·29-s + 10·31-s − 4·41-s − 2·49-s − 20·55-s − 8·59-s + 16·61-s + 12·71-s + 18·79-s + 28·89-s + 24·95-s + 34·101-s + 16·109-s + 53·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 36·145-s + 149-s + 151-s − 20·155-s + ⋯
L(s)  = 1  − 0.894·5-s + 3.01·11-s − 2.75·19-s − 1/5·25-s + 3.34·29-s + 1.79·31-s − 0.624·41-s − 2/7·49-s − 2.69·55-s − 1.04·59-s + 2.04·61-s + 1.42·71-s + 2.02·79-s + 2.96·89-s + 2.46·95-s + 3.38·101-s + 1.53·109-s + 4.81·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.98·145-s + 0.0819·149-s + 0.0813·151-s − 1.60·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4665600\)    =    \(2^{8} \cdot 3^{6} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(297.482\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4665600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.765765129\)
\(L(\frac12)\) \(\approx\) \(2.765765129\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.13.a_ar
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.17.a_abh
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.23.a_abt
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.29.as_fj
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.43.a_adh
47$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \) 2.47.a_cx
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.79.as_jf
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−8.976900416331365502461194250485, −8.935287782589884645318706684027, −8.466009955994635674350779298181, −8.232303287040387939575464283783, −7.912474994140581120583008461352, −7.24576276594186805667364826961, −6.63261542824311388824443934446, −6.49753885714853443022114435659, −6.39814711543071832995400608678, −6.10704466593891491551517401134, −5.05789318226404338438379554810, −4.61463244384163357566221075484, −4.38595006375639215852257042785, −4.10565520611227391321923074628, −3.42089892076631008528000220231, −3.34711338459262275031703629110, −2.22551931996650733119819391832, −2.08790520080880437263343051638, −1.02952460791105264657679749735, −0.73992548487038741356440557818, 0.73992548487038741356440557818, 1.02952460791105264657679749735, 2.08790520080880437263343051638, 2.22551931996650733119819391832, 3.34711338459262275031703629110, 3.42089892076631008528000220231, 4.10565520611227391321923074628, 4.38595006375639215852257042785, 4.61463244384163357566221075484, 5.05789318226404338438379554810, 6.10704466593891491551517401134, 6.39814711543071832995400608678, 6.49753885714853443022114435659, 6.63261542824311388824443934446, 7.24576276594186805667364826961, 7.912474994140581120583008461352, 8.232303287040387939575464283783, 8.466009955994635674350779298181, 8.935287782589884645318706684027, 8.976900416331365502461194250485

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