Properties

Label 4-2200e2-1.1-c1e2-0-12
Degree $4$
Conductor $4840000$
Sign $-1$
Analytic cond. $308.602$
Root an. cond. $4.19131$
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 − 2·9-s + 2·11-s − 2·23-s − 10·27-s − 4·31-s + 4·33-s + 8·37-s − 6·47-s + 10·49-s − 4·53-s − 8·59-s − 10·67-s − 4·69-s + 4·71-s − 5·81-s + 4·89-s − 8·93-s − 4·99-s + 30·103-s + 16·111-s − 7·121-s + ⋯
L(s)  = 1  + 1.15·3-s − 2/3·9-s + 0.603·11-s − 0.417·23-s − 1.92·27-s − 0.718·31-s + 0.696·33-s + 1.31·37-s − 0.875·47-s + 10/7·49-s − 0.549·53-s − 1.04·59-s − 1.22·67-s − 0.481·69-s + 0.474·71-s − 5/9·81-s + 0.423·89-s − 0.829·93-s − 0.402·99-s + 2.95·103-s + 1.51·111-s − 0.636·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4840000\)    =    \(2^{6} \cdot 5^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(308.602\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4840000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.3.ac_g
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.c_bm
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ai_cc
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.43.a_da
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.g_dq
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.i_eo
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.k_da
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ae_fm
73$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.73.a_aba
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.83.a_abi
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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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

−7.39705941478887453240568671647, −6.71971185891233109868222381385, −6.25677523343486614807276599772, −5.90832418893146564430994023317, −5.68186525456105717305879822197, −4.91172548705591846281859809045, −4.66821301561363437137110301188, −3.93548024987083247436715813678, −3.72425951702123901157875792385, −3.12261187286125204328679080504, −2.84475270865979962490794889644, −2.21978968608376499479777271773, −1.84125821762031174802420431519, −1.00373087624558112290875301584, 0, 1.00373087624558112290875301584, 1.84125821762031174802420431519, 2.21978968608376499479777271773, 2.84475270865979962490794889644, 3.12261187286125204328679080504, 3.72425951702123901157875792385, 3.93548024987083247436715813678, 4.66821301561363437137110301188, 4.91172548705591846281859809045, 5.68186525456105717305879822197, 5.90832418893146564430994023317, 6.25677523343486614807276599772, 6.71971185891233109868222381385, 7.39705941478887453240568671647

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