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Properties

Label 4-5227200-1.1-c1e2-0-88
Degree $4$
Conductor $5227200$
Sign $1$
Analytic cond. $333.290$
Root an. cond. $4.27273$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s + 2·15-s − 8·23-s − 25-s − 27-s − 16·31-s − 2·45-s − 10·49-s − 12·53-s + 8·69-s + 75-s + 81-s + 16·93-s + 12·113-s + 16·115-s − 11·121-s + 12·125-s + 127-s + 131-s + 2·135-s + 137-s + 139-s + 10·147-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.516·15-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 2.87·31-s − 0.298·45-s − 1.42·49-s − 1.64·53-s + 0.963·69-s + 0.115·75-s + 1/9·81-s + 1.65·93-s + 1.12·113-s + 1.49·115-s − 121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.172·135-s + 0.0854·137-s + 0.0848·139-s + 0.824·147-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5227200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5227200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5227200\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(333.290\)
Root analytic conductor: \(4.27273\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5227200,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( 1 + T \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.61.a_aec
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.a_fa
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−7.08500531155412670566435678678, −6.40915952946768264822130690853, −6.03764361909688342972641306478, −5.75906999737801844177573176861, −5.27009716350970270309843990979, −4.75018503107772330327450209875, −4.45476090548055481383350436983, −3.83634494530242736724715515484, −3.58371050852763691874559475037, −3.21739310344995440343846499552, −2.27966056245936996521038374463, −1.88489067883337711859690167958, −1.26609681159415606131405037588, 0, 0, 1.26609681159415606131405037588, 1.88489067883337711859690167958, 2.27966056245936996521038374463, 3.21739310344995440343846499552, 3.58371050852763691874559475037, 3.83634494530242736724715515484, 4.45476090548055481383350436983, 4.75018503107772330327450209875, 5.27009716350970270309843990979, 5.75906999737801844177573176861, 6.03764361909688342972641306478, 6.40915952946768264822130690853, 7.08500531155412670566435678678

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