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Properties

Label 4-1152e2-1.1-c3e2-0-10
Degree 44
Conductor 13271041327104
Sign 11
Analytic cond. 4619.944619.94
Root an. cond. 8.244408.24440
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  − 64·7-s + 196·17-s − 64·23-s + 106·25-s − 512·31-s + 204·41-s + 640·47-s + 2.38e3·49-s + 832·71-s − 276·73-s − 128·79-s − 1.16e3·89-s + 476·97-s + 1.98e3·103-s + 604·113-s − 1.25e4·119-s + 2.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4.09e3·161-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.45·7-s + 2.79·17-s − 0.580·23-s + 0.847·25-s − 2.96·31-s + 0.777·41-s + 1.98·47-s + 6.95·49-s + 1.39·71-s − 0.442·73-s − 0.182·79-s − 1.38·89-s + 0.498·97-s + 1.89·103-s + 0.502·113-s − 9.66·119-s + 1.95·121-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 + 0.000508·157-s + 2.00·161-s + 0.000480·163-s + 0.000463·167-s + ⋯

Functional equation

Λ(s)=(1327104s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1327104s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 13271041327104    =    214⋅342^{14} \cdot 3^{4}
Sign: 11
Analytic conductor: 4619.944619.94
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1327104, ( :3/2,3/2), 1)(4,\ 1327104,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.8301199291.830119929
L(12)L(\frac12) ≈\approx 1.8301199291.830119929
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
3 1 1
good5C22C_2^2 1−106T2+p6T4 1 - 106 T^{2} + p^{6} T^{4}
7C2C_2 (1+32T+p3T2)2 ( 1 + 32 T + p^{3} T^{2} )^{2}
11C22C_2^2 1−2598T2+p6T4 1 - 2598 T^{2} + p^{6} T^{4}
13C22C_2^2 1−3994T2+p6T4 1 - 3994 T^{2} + p^{6} T^{4}
17C2C_2 (1−98T+p3T2)2 ( 1 - 98 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−5974T2+p6T4 1 - 5974 T^{2} + p^{6} T^{4}
23C2C_2 (1+32T+p3T2)2 ( 1 + 32 T + p^{3} T^{2} )^{2}
29C22C_2^2 1−19194T2+p6T4 1 - 19194 T^{2} + p^{6} T^{4}
31C2C_2 (1+256T+p3T2)2 ( 1 + 256 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−92842T2+p6T4 1 - 92842 T^{2} + p^{6} T^{4}
41C2C_2 (1−102T+p3T2)2 ( 1 - 102 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−71398T2+p6T4 1 - 71398 T^{2} + p^{6} T^{4}
47C2C_2 (1−320T+p3T2)2 ( 1 - 320 T + p^{3} T^{2} )^{2}
53C22C_2^2 1−291978T2+p6T4 1 - 291978 T^{2} + p^{6} T^{4}
59C22C_2^2 1−244294T2+p6T4 1 - 244294 T^{2} + p^{6} T^{4}
61C22C_2^2 1−49466T2+p6T4 1 - 49466 T^{2} + p^{6} T^{4}
67C22C_2^2 1−296822T2+p6T4 1 - 296822 T^{2} + p^{6} T^{4}
71C2C_2 (1−416T+p3T2)2 ( 1 - 416 T + p^{3} T^{2} )^{2}
73C2C_2 (1+138T+p3T2)2 ( 1 + 138 T + p^{3} T^{2} )^{2}
79C2C_2 (1+64T+p3T2)2 ( 1 + 64 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−989910T2+p6T4 1 - 989910 T^{2} + p^{6} T^{4}
89C2C_2 (1+582T+p3T2)2 ( 1 + 582 T + p^{3} T^{2} )^{2}
97C2C_2 (1−238T+p3T2)2 ( 1 - 238 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

−9.666664658809018361517647869113, −9.392656189466090410047124048107, −8.890809329735441799240492357002, −8.674862319785661612836621078538, −7.66418723613323904447223764772, −7.58914452445000718275572054090, −7.07104639519245952422796475666, −6.79746673901580801844728352654, −6.15714614460348207807664741314, −5.85967078359538650728688935205, −5.60699025300210885792316331158, −5.17957456048342191021504473275, −4.00312819470748420122643950859, −3.88892057072972161905692027248, −3.25915613963670719844321856028, −3.15367373787743586084644408835, −2.59077756564567735159746202525, −1.74096922812398386298285286014, −0.67341887266093627643859601061, −0.52020198516688606631277704698, 0.52020198516688606631277704698, 0.67341887266093627643859601061, 1.74096922812398386298285286014, 2.59077756564567735159746202525, 3.15367373787743586084644408835, 3.25915613963670719844321856028, 3.88892057072972161905692027248, 4.00312819470748420122643950859, 5.17957456048342191021504473275, 5.60699025300210885792316331158, 5.85967078359538650728688935205, 6.15714614460348207807664741314, 6.79746673901580801844728352654, 7.07104639519245952422796475666, 7.58914452445000718275572054090, 7.66418723613323904447223764772, 8.674862319785661612836621078538, 8.890809329735441799240492357002, 9.392656189466090410047124048107, 9.666664658809018361517647869113

Graph of the ZZ-function along the critical line