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Properties

Label 4-5227200-1.1-c1e2-0-66
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 4·11-s + 4·23-s − 5·25-s − 27-s + 8·31-s − 4·33-s + 2·37-s − 16·47-s − 4·49-s − 2·53-s + 16·59-s − 4·69-s − 24·71-s + 5·75-s + 81-s + 4·89-s − 8·93-s + 14·97-s + 4·99-s − 20·103-s − 2·111-s − 10·113-s + 5·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.834·23-s − 25-s − 0.192·27-s + 1.43·31-s − 0.696·33-s + 0.328·37-s − 2.33·47-s − 4/7·49-s − 0.274·53-s + 2.08·59-s − 0.481·69-s − 2.84·71-s + 0.577·75-s + 1/9·81-s + 0.423·89-s − 0.829·93-s + 1.42·97-s + 0.402·99-s − 1.97·103-s − 0.189·111-s − 0.940·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-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: \(1\)
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$C_2$ \( 1 + p T^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
13$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.13.a_q
17$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.17.a_ae
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ae_bu
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ac_co
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.43.a_ae
47$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.q_fm
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.c_ao
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aq_gk
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.61.a_k
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2^2$ \( 1 + 136 T^{2} + p^{2} T^{4} \) 2.73.a_fg
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.83.a_ea
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.ae_de
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ao_es
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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

−6.92318812947709829857199933577, −6.59060614834756228025102481824, −6.41176650308245350999104202909, −5.95789367429028619689585123868, −5.43620010668972877659421102935, −5.09288483700304703580687720384, −4.53661157539686235655561042088, −4.26483165984210633367059547512, −3.77279415780071677436228264553, −3.23066597332443980701345432846, −2.80106838852475612540391021742, −2.05819683239581432452136057540, −1.45896191837315043467382601463, −0.991408300446891656385657480292, 0, 0.991408300446891656385657480292, 1.45896191837315043467382601463, 2.05819683239581432452136057540, 2.80106838852475612540391021742, 3.23066597332443980701345432846, 3.77279415780071677436228264553, 4.26483165984210633367059547512, 4.53661157539686235655561042088, 5.09288483700304703580687720384, 5.43620010668972877659421102935, 5.95789367429028619689585123868, 6.41176650308245350999104202909, 6.59060614834756228025102481824, 6.92318812947709829857199933577

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