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Properties

Label 4-1584e2-1.1-c1e2-0-37
Degree $4$
Conductor $2509056$
Sign $-1$
Analytic cond. $159.979$
Root an. cond. $3.55644$
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  − 2·11-s − 6·19-s + 4·25-s + 8·41-s + 2·43-s − 12·49-s + 8·59-s − 20·67-s − 16·73-s + 16·83-s + 14·89-s + 8·97-s − 16·107-s + 14·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 0.603·11-s − 1.37·19-s + 4/5·25-s + 1.24·41-s + 0.304·43-s − 1.71·49-s + 1.04·59-s − 2.44·67-s − 1.87·73-s + 1.75·83-s + 1.48·89-s + 0.812·97-s − 1.54·107-s + 1.31·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.23·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2509056\)    =    \(2^{8} \cdot 3^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(159.979\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2509056,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
13$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.13.a_aq
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.g_bu
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.23.a_au
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.37.a_abe
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ac_ck
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.53.a_ca
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ai_eo
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.61.a_abo
67$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.u_iw
71$C_2^2$ \( 1 - 84 T^{2} + p^{2} T^{4} \) 2.71.a_adg
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.q_gs
79$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \) 2.79.a_do
83$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.aq_is
89$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.ao_is
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.ai_co
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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

−7.40695915479543820062330332306, −7.08227321668560425260587693841, −6.53989413744638658993708311811, −6.15698692100555494138475391574, −5.85868782808563230942383253464, −5.27749228664637173942242287885, −4.76427801847053228922208112018, −4.46195367601115603381843732857, −4.01525660380416543815215920790, −3.27984240827439812578270523146, −2.94990683722156260057326279560, −2.28768351638594252541192101144, −1.82993020077928243366348330483, −0.950317352835481520599939452040, 0, 0.950317352835481520599939452040, 1.82993020077928243366348330483, 2.28768351638594252541192101144, 2.94990683722156260057326279560, 3.27984240827439812578270523146, 4.01525660380416543815215920790, 4.46195367601115603381843732857, 4.76427801847053228922208112018, 5.27749228664637173942242287885, 5.85868782808563230942383253464, 6.15698692100555494138475391574, 6.53989413744638658993708311811, 7.08227321668560425260587693841, 7.40695915479543820062330332306

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