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Properties

Label 4-1280e2-1.1-c3e2-0-12
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 5703.635703.63
Root an. cond. 8.690368.69036
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·7-s + 50·9-s + 52·17-s + 156·23-s − 25·25-s − 216·31-s − 44·41-s − 1.02e3·47-s − 578·49-s − 600·63-s − 824·71-s + 1.75e3·73-s + 1.20e3·79-s + 1.77e3·81-s + 300·89-s + 772·97-s + 1.19e3·103-s + 3.12e3·113-s − 624·119-s + 1.63e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.60e3·153-s + ⋯
L(s)  = 1  − 0.647·7-s + 1.85·9-s + 0.741·17-s + 1.41·23-s − 1/5·25-s − 1.25·31-s − 0.167·41-s − 3.19·47-s − 1.68·49-s − 1.19·63-s − 1.37·71-s + 2.81·73-s + 1.70·79-s + 2.42·81-s + 0.357·89-s + 0.808·97-s + 1.14·103-s + 2.60·113-s − 0.480·119-s + 1.23·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 1.37·153-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 5703.635703.63
Root analytic conductor: 8.690368.69036
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :3/2,3/2), 1)(4,\ 1638400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.3130811293.313081129
L(12)L(\frac12) ≈\approx 3.3130811293.313081129
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
5C2C_2 1+p2T2 1 + p^{2} T^{2}
good3C22C_2^2 1−50T2+p6T4 1 - 50 T^{2} + p^{6} T^{4}
7C2C_2 (1+6T+p3T2)2 ( 1 + 6 T + p^{3} T^{2} )^{2}
11C22C_2^2 1−1638T2+p6T4 1 - 1638 T^{2} + p^{6} T^{4}
13C22C_2^2 1−2950T2+p6T4 1 - 2950 T^{2} + p^{6} T^{4}
17C2C_2 (1−26T+p3T2)2 ( 1 - 26 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−3718T2+p6T4 1 - 3718 T^{2} + p^{6} T^{4}
23C2C_2 (1−78T+p3T2)2 ( 1 - 78 T + p^{3} T^{2} )^{2}
29C22C_2^2 1−46278T2+p6T4 1 - 46278 T^{2} + p^{6} T^{4}
31C2C_2 (1+108T+p3T2)2 ( 1 + 108 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−30550T2+p6T4 1 - 30550 T^{2} + p^{6} T^{4}
41C2C_2 (1+22T+p3T2)2 ( 1 + 22 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+36350T2+p6T4 1 + 36350 T^{2} + p^{6} T^{4}
47C2C_2 (1+514T+p3T2)2 ( 1 + 514 T + p^{3} T^{2} )^{2}
53C22C_2^2 1−297750T2+p6T4 1 - 297750 T^{2} + p^{6} T^{4}
59C22C_2^2 1−160758T2+p6T4 1 - 160758 T^{2} + p^{6} T^{4}
61C22C_2^2 1−185638T2+p6T4 1 - 185638 T^{2} + p^{6} T^{4}
67C22C_2^2 1−585650T2+p6T4 1 - 585650 T^{2} + p^{6} T^{4}
71C2C_2 (1+412T+p3T2)2 ( 1 + 412 T + p^{3} T^{2} )^{2}
73C2C_2 (1−878T+p3T2)2 ( 1 - 878 T + p^{3} T^{2} )^{2}
79C2C_2 (1−600T+p3T2)2 ( 1 - 600 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−1064050T2+p6T4 1 - 1064050 T^{2} + p^{6} T^{4}
89C2C_2 (1−150T+p3T2)2 ( 1 - 150 T + p^{3} T^{2} )^{2}
97C2C_2 (1−386T+p3T2)2 ( 1 - 386 T + p^{3} T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(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

−9.502840790716678306870017170525, −9.379882434636669955351346149267, −8.648867879379876927101683888800, −8.324399933382693724596262214049, −7.61063701121957513711166948380, −7.60701935564063159120556809079, −6.95888381870986730612657390245, −6.70782261965001786869428689032, −6.31868363154200435459794213377, −5.82575371851561993031743552373, −5.03053531508750380963378540030, −4.91936563217609242711024672185, −4.48623018539530115482068681255, −3.70643185142289997713092646597, −3.30123780713131655900810561388, −3.19947598367517014474693879066, −1.94573033343062993260221654129, −1.84648132656004475961690105100, −1.04467332713503474572271910760, −0.47795396462831431387301314890, 0.47795396462831431387301314890, 1.04467332713503474572271910760, 1.84648132656004475961690105100, 1.94573033343062993260221654129, 3.19947598367517014474693879066, 3.30123780713131655900810561388, 3.70643185142289997713092646597, 4.48623018539530115482068681255, 4.91936563217609242711024672185, 5.03053531508750380963378540030, 5.82575371851561993031743552373, 6.31868363154200435459794213377, 6.70782261965001786869428689032, 6.95888381870986730612657390245, 7.60701935564063159120556809079, 7.61063701121957513711166948380, 8.324399933382693724596262214049, 8.648867879379876927101683888800, 9.379882434636669955351346149267, 9.502840790716678306870017170525

Graph of the ZZ-function along the critical line