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Properties

Label 4-3744e2-1.1-c1e2-0-18
Degree 44
Conductor 1401753614017536
Sign 11
Analytic cond. 893.770893.770
Root an. cond. 5.467725.46772
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 3·7-s + 4·11-s − 2·13-s − 3·17-s − 12·19-s + 25-s − 6·31-s − 9·35-s − 7·37-s − 6·41-s − 3·43-s − 9·47-s − 3·49-s + 18·53-s + 12·55-s − 12·59-s + 2·61-s − 6·65-s − 12·67-s + 9·71-s + 20·73-s − 12·77-s − 24·79-s − 10·83-s − 9·85-s + 6·91-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.13·7-s + 1.20·11-s − 0.554·13-s − 0.727·17-s − 2.75·19-s + 1/5·25-s − 1.07·31-s − 1.52·35-s − 1.15·37-s − 0.937·41-s − 0.457·43-s − 1.31·47-s − 3/7·49-s + 2.47·53-s + 1.61·55-s − 1.56·59-s + 0.256·61-s − 0.744·65-s − 1.46·67-s + 1.06·71-s + 2.34·73-s − 1.36·77-s − 2.70·79-s − 1.09·83-s − 0.976·85-s + 0.628·91-s + ⋯

Functional equation

Λ(s)=(14017536s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(14017536s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1401753614017536    =    210⋅34⋅1322^{10} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 893.770893.770
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 14017536, ( :1/2,1/2), 1)(4,\ 14017536,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
13C1C_1 (1+T)2 ( 1 + T )^{2}
good5C22C_2^2 1−3T+8T2−3pT3+p2T4 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4}
7D4D_{4} 1+3T+12T2+3pT3+p2T4 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17D4D_{4} 1+3T+32T2+3pT3+p2T4 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4}
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31D4D_{4} 1+6T+54T2+6pT3+p2T4 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+7T+48T2+7pT3+p2T4 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+74T2+6pT3+p2T4 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+3T−18T2+3pT3+p2T4 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+9T+76T2+9pT3+p2T4 1 + 9 T + 76 T^{2} + 9 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−18T+170T2−18pT3+p2T4 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4}
59C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
61D4D_{4} 1−2T−30T2−2pT3+p2T4 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4}
67C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
71D4D_{4} 1−9T+124T2−9pT3+p2T4 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
83D4D_{4} 1+10T+38T2+10pT3+p2T4 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p 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

−8.359247470924614125970790125691, −8.310080589934033374812375342089, −7.33713268528036076349461049532, −7.09046374063523696582329303791, −6.74069179733242319110287892029, −6.45394742968289184001431464816, −6.16318255530587707759055538034, −5.94728239294354965613847389707, −5.28505144718888428024789098250, −5.08379743035954149505590828294, −4.30570908196938274634522827852, −4.23642977810300404332541306626, −3.58569493556401252252580430744, −3.33471575680941272079501992305, −2.43470077497720609517368412384, −2.38466120869572987838380453888, −1.67036395773389906988686940996, −1.49264770548250293594637876598, 0, 0, 1.49264770548250293594637876598, 1.67036395773389906988686940996, 2.38466120869572987838380453888, 2.43470077497720609517368412384, 3.33471575680941272079501992305, 3.58569493556401252252580430744, 4.23642977810300404332541306626, 4.30570908196938274634522827852, 5.08379743035954149505590828294, 5.28505144718888428024789098250, 5.94728239294354965613847389707, 6.16318255530587707759055538034, 6.45394742968289184001431464816, 6.74069179733242319110287892029, 7.09046374063523696582329303791, 7.33713268528036076349461049532, 8.310080589934033374812375342089, 8.359247470924614125970790125691

Graph of the ZZ-function along the critical line