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Properties

Label 4-2496e2-1.1-c1e2-0-2
Degree $4$
Conductor $6230016$
Sign $1$
Analytic cond. $397.231$
Root an. cond. $4.46437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 2·13-s + 12·23-s + 8·25-s + 4·27-s − 4·37-s + 4·39-s − 24·47-s − 4·49-s − 16·61-s − 24·69-s − 24·71-s + 4·73-s − 16·75-s − 11·81-s + 24·83-s − 20·97-s − 24·107-s − 4·109-s + 8·111-s − 2·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.554·13-s + 2.50·23-s + 8/5·25-s + 0.769·27-s − 0.657·37-s + 0.640·39-s − 3.50·47-s − 4/7·49-s − 2.04·61-s − 2.88·69-s − 2.84·71-s + 0.468·73-s − 1.84·75-s − 1.22·81-s + 2.63·83-s − 2.03·97-s − 2.32·107-s − 0.383·109-s + 0.759·111-s − 0.184·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6230016\)    =    \(2^{12} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(397.231\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6230016,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7157410684\)
\(L(\frac12)\) \(\approx\) \(0.7157410684\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.31.a_abs
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.41.a_adc
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \) 2.67.a_aem
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.79.a_adi
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \) 2.89.a_aey
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−9.115927017210107657419112042440, −8.839088466692589603269792597626, −8.359434629217760221147641005071, −7.920460628607853711662591199969, −7.55148289238900479761853146416, −6.96948567783300939658450681801, −6.79676077873873158056216667873, −6.42372197197166788844178362495, −6.17844835843997185499768170738, −5.34876883963515057545429861033, −5.23117870242481600587779957832, −4.81357870413969561667446293811, −4.70066714191072884606691469290, −3.97426791652236557253659632781, −3.17779129694635953876009540049, −3.03500125608773468115022531176, −2.59951624081907006934421405271, −1.41819956229099312237092660478, −1.38065508868933377644396312672, −0.32592372835047223712539166839, 0.32592372835047223712539166839, 1.38065508868933377644396312672, 1.41819956229099312237092660478, 2.59951624081907006934421405271, 3.03500125608773468115022531176, 3.17779129694635953876009540049, 3.97426791652236557253659632781, 4.70066714191072884606691469290, 4.81357870413969561667446293811, 5.23117870242481600587779957832, 5.34876883963515057545429861033, 6.17844835843997185499768170738, 6.42372197197166788844178362495, 6.79676077873873158056216667873, 6.96948567783300939658450681801, 7.55148289238900479761853146416, 7.920460628607853711662591199969, 8.359434629217760221147641005071, 8.839088466692589603269792597626, 9.115927017210107657419112042440

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