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Properties

Label 4-387200-1.1-c1e2-0-23
Degree $4$
Conductor $387200$
Sign $1$
Analytic cond. $24.6882$
Root an. cond. $2.22906$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 6·9-s − 8·13-s − 8·17-s + 3·25-s − 12·29-s − 4·37-s + 12·41-s + 12·45-s − 10·49-s − 20·53-s − 12·61-s + 16·65-s + 8·73-s + 27·81-s + 16·85-s + 12·89-s − 4·97-s + 12·101-s − 20·109-s + 28·113-s + 48·117-s + 121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 2·9-s − 2.21·13-s − 1.94·17-s + 3/5·25-s − 2.22·29-s − 0.657·37-s + 1.87·41-s + 1.78·45-s − 1.42·49-s − 2.74·53-s − 1.53·61-s + 1.98·65-s + 0.936·73-s + 3·81-s + 1.73·85-s + 1.27·89-s − 0.406·97-s + 1.19·101-s − 1.91·109-s + 2.63·113-s + 4.43·117-s + 1/11·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(387200\)    =    \(2^{7} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(24.6882\)
Root analytic conductor: \(2.22906\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 387200,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.43.a_de
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.83.a_gg
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−7.978734464208958320503765844199, −7.906393321228809348111883565536, −7.36247134612527590927307037370, −6.94257154461568951821487101757, −6.21087114261569312686272580596, −5.99500771172096974750229105298, −5.17000472187588479255187728599, −4.82772136865313429170223708877, −4.45821261643777188910209572887, −3.58711967186630926697460909520, −3.12595338350487188300141328594, −2.44118808260323168492313335198, −2.03399759604601396643465282622, 0, 0, 2.03399759604601396643465282622, 2.44118808260323168492313335198, 3.12595338350487188300141328594, 3.58711967186630926697460909520, 4.45821261643777188910209572887, 4.82772136865313429170223708877, 5.17000472187588479255187728599, 5.99500771172096974750229105298, 6.21087114261569312686272580596, 6.94257154461568951821487101757, 7.36247134612527590927307037370, 7.906393321228809348111883565536, 7.978734464208958320503765844199

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