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Properties

Label 4-3960e2-1.1-c1e2-0-7
Degree $4$
Conductor $15681600$
Sign $1$
Analytic cond. $999.872$
Root an. cond. $5.62323$
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  + 2·5-s + 2·11-s − 8·19-s − 25-s + 4·29-s + 16·31-s + 12·41-s + 10·49-s + 4·55-s + 8·59-s + 28·61-s − 16·71-s + 16·79-s − 20·89-s − 16·95-s + 20·101-s + 28·109-s + 3·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 32·155-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s − 1.83·19-s − 1/5·25-s + 0.742·29-s + 2.87·31-s + 1.87·41-s + 10/7·49-s + 0.539·55-s + 1.04·59-s + 3.58·61-s − 1.89·71-s + 1.80·79-s − 2.11·89-s − 1.64·95-s + 1.99·101-s + 2.68·109-s + 3/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15681600\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(999.872\)
Root analytic conductor: \(5.62323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15681600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.262624915\)
\(L(\frac12)\) \(\approx\) \(4.262624915\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.47.a_adm
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \) 2.83.a_agg
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−8.561690076916054853886704956183, −8.509151204991037720525387181518, −7.969278943331290246984175117134, −7.56380120870672226378041199914, −7.07471794438814770083035953668, −6.68899086485885057007120434305, −6.39804456224879840857164182770, −6.17149480136203216358665343108, −5.66882303551193056114112541739, −5.48925641320362647852716259597, −4.67497077699503002441332596121, −4.58221927067700631649930411133, −4.02440603694632319159604924134, −3.86762120371455746166277772484, −3.00571192406293325630737643907, −2.65791307822691878667918729706, −2.15097134920176642783220814059, −1.95780733126578720706026041785, −0.913959038410848158142972276161, −0.76682447496435372864109877362, 0.76682447496435372864109877362, 0.913959038410848158142972276161, 1.95780733126578720706026041785, 2.15097134920176642783220814059, 2.65791307822691878667918729706, 3.00571192406293325630737643907, 3.86762120371455746166277772484, 4.02440603694632319159604924134, 4.58221927067700631649930411133, 4.67497077699503002441332596121, 5.48925641320362647852716259597, 5.66882303551193056114112541739, 6.17149480136203216358665343108, 6.39804456224879840857164182770, 6.68899086485885057007120434305, 7.07471794438814770083035953668, 7.56380120870672226378041199914, 7.969278943331290246984175117134, 8.509151204991037720525387181518, 8.561690076916054853886704956183

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