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Properties

Label 4-2624e2-1.1-c1e2-0-12
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·9-s + 2·25-s − 16·31-s − 4·37-s − 6·41-s − 8·43-s + 8·45-s + 6·49-s − 8·59-s + 4·61-s + 28·73-s − 5·81-s + 24·83-s − 16·103-s + 8·107-s + 12·113-s + 14·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·155-s + 157-s + ⋯
L(s)  = 1  − 1.78·5-s − 2/3·9-s + 2/5·25-s − 2.87·31-s − 0.657·37-s − 0.937·41-s − 1.21·43-s + 1.19·45-s + 6/7·49-s − 1.04·59-s + 0.512·61-s + 3.27·73-s − 5/9·81-s + 2.63·83-s − 1.57·103-s + 0.773·107-s + 1.12·113-s + 1.27·121-s + 2.50·125-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 + 5.14·155-s + 0.0798·157-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: \(2\)
Selberg data: \((4,\ 6885376,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
41$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
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.q_ew
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.47.a_adi
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.71.a_acs
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.79.a_afu
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.89.a_aby
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−8.888843940649523537033929649213, −8.135718645356398044431926469390, −8.061288689564869938290171219696, −7.53048881109507139472483898254, −7.19329389521508500902345398463, −7.03430143165493489831648797664, −6.29784127022138595739276887929, −6.11116830385895361147868668936, −5.38689063119039160929432432020, −5.15596710736020945436048426702, −4.82694579010484951729055065039, −4.05288902500552237793673681572, −3.83013743400294434522060536194, −3.34658419763100065290472210068, −3.34203886798213902015147284516, −2.22885376261039609154321513075, −2.03754132794151406483654831090, −1.09674807355815801470750320281, 0, 0, 1.09674807355815801470750320281, 2.03754132794151406483654831090, 2.22885376261039609154321513075, 3.34203886798213902015147284516, 3.34658419763100065290472210068, 3.83013743400294434522060536194, 4.05288902500552237793673681572, 4.82694579010484951729055065039, 5.15596710736020945436048426702, 5.38689063119039160929432432020, 6.11116830385895361147868668936, 6.29784127022138595739276887929, 7.03430143165493489831648797664, 7.19329389521508500902345398463, 7.53048881109507139472483898254, 8.061288689564869938290171219696, 8.135718645356398044431926469390, 8.888843940649523537033929649213

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