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Properties

Label 4-4080e2-1.1-c1e2-0-16
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
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  + 2·5-s − 9-s + 6·11-s + 6·19-s − 25-s − 18·29-s + 20·31-s + 18·41-s − 2·45-s − 11·49-s + 12·55-s − 4·59-s − 16·61-s + 28·79-s + 81-s − 20·89-s + 12·95-s − 6·99-s + 24·101-s − 8·109-s + 5·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 36·145-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s + 1.80·11-s + 1.37·19-s − 1/5·25-s − 3.34·29-s + 3.59·31-s + 2.81·41-s − 0.298·45-s − 1.57·49-s + 1.61·55-s − 0.520·59-s − 2.04·61-s + 3.15·79-s + 1/9·81-s − 2.11·89-s + 1.23·95-s − 0.603·99-s + 2.38·101-s − 0.766·109-s + 5/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.98·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.087932490\)
\(L(\frac12)\) \(\approx\) \(4.087932490\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.37.a_abx
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.41.as_gh
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.47.a_an
53$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \) 2.53.a_acf
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.59.e_es
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \) 2.73.a_aer
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−8.892965752643198093879885738717, −8.073419065897992442593698920991, −7.80920598877544450684320066358, −7.68890786552289727063054964526, −7.16158539317533809619384532199, −6.61632785647335872713414092995, −6.34336516413846296818840958321, −6.02521393351708228044451450252, −5.87627618091990320965875753314, −5.27462872895707065078393488640, −4.92555605849474306627448735944, −4.45560089338045654810322872221, −3.92756432202477772802475673772, −3.77389459555201031357691930769, −3.05042532205806082692255862442, −2.80263971952542366192847975899, −2.16288424527309558308191178347, −1.61503053817501359258444373248, −1.24908488843468204699211872750, −0.61566727083096377946701573493, 0.61566727083096377946701573493, 1.24908488843468204699211872750, 1.61503053817501359258444373248, 2.16288424527309558308191178347, 2.80263971952542366192847975899, 3.05042532205806082692255862442, 3.77389459555201031357691930769, 3.92756432202477772802475673772, 4.45560089338045654810322872221, 4.92555605849474306627448735944, 5.27462872895707065078393488640, 5.87627618091990320965875753314, 6.02521393351708228044451450252, 6.34336516413846296818840958321, 6.61632785647335872713414092995, 7.16158539317533809619384532199, 7.68890786552289727063054964526, 7.80920598877544450684320066358, 8.073419065897992442593698920991, 8.892965752643198093879885738717

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