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Properties

Label 4-2624e2-1.1-c1e2-0-5
Degree $4$
Conductor $6885376$
Sign $1$
Analytic cond. $439.017$
Root an. cond. $4.57741$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s − 10·25-s + 16·31-s − 16·37-s − 6·41-s + 8·43-s − 4·49-s + 24·59-s − 4·61-s − 8·73-s + 7·81-s − 24·83-s − 32·103-s + 24·107-s + 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯
L(s)  = 1  + 4/3·9-s − 2·25-s + 2.87·31-s − 2.63·37-s − 0.937·41-s + 1.21·43-s − 4/7·49-s + 3.12·59-s − 0.512·61-s − 0.936·73-s + 7/9·81-s − 2.63·83-s − 3.15·103-s + 2.32·107-s + 2.25·113-s + 1.81·121-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 + 2·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6885376\)    =    \(2^{12} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(439.017\)
Root analytic conductor: \(4.57741\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6885376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.399649674\)
\(L(\frac12)\) \(\approx\) \(2.399649674\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.47.a_abs
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.67.a_bc
71$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.71.a_ado
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.79.a_afk
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
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

−8.891469499644162543686393627221, −8.711935362436529781211727129097, −8.225523209629705612102174446168, −8.037667306750848430173670108986, −7.44792138415604499619483341702, −7.16115888324212279685532985558, −6.75673217460874412592094415942, −6.56200657441389535957500927101, −5.94128958397620252483877214123, −5.52506245666531720333649824480, −5.26449181306628346759390542488, −4.51447888170071713116336498674, −4.33683833760827064277083220834, −4.00861174572670159890331347180, −3.36671125854621519361532776409, −3.00453203648851276865586032690, −2.25275594701748714925847099885, −1.83695266189739431821786959799, −1.31413137459246018953521091165, −0.52931597470897957209954358229, 0.52931597470897957209954358229, 1.31413137459246018953521091165, 1.83695266189739431821786959799, 2.25275594701748714925847099885, 3.00453203648851276865586032690, 3.36671125854621519361532776409, 4.00861174572670159890331347180, 4.33683833760827064277083220834, 4.51447888170071713116336498674, 5.26449181306628346759390542488, 5.52506245666531720333649824480, 5.94128958397620252483877214123, 6.56200657441389535957500927101, 6.75673217460874412592094415942, 7.16115888324212279685532985558, 7.44792138415604499619483341702, 8.037667306750848430173670108986, 8.225523209629705612102174446168, 8.711935362436529781211727129097, 8.891469499644162543686393627221

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