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Properties

Label 2-5376-1.1-c1-0-44
Degree 22
Conductor 53765376
Sign 11
Analytic cond. 42.927542.9275
Root an. cond. 6.551916.55191
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2.44·5-s − 7-s + 9-s + 0.449·11-s + 6.89·13-s − 2.44·15-s + 1.55·17-s + 6.89·19-s + 21-s + 4.44·23-s + 0.999·25-s − 27-s + 6.89·29-s + 2.89·31-s − 0.449·33-s − 2.44·35-s − 2·37-s − 6.89·39-s + 7.34·41-s − 12.8·43-s + 2.44·45-s − 8.89·47-s + 49-s − 1.55·51-s − 10·53-s + 1.10·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.09·5-s − 0.377·7-s + 0.333·9-s + 0.135·11-s + 1.91·13-s − 0.632·15-s + 0.376·17-s + 1.58·19-s + 0.218·21-s + 0.927·23-s + 0.199·25-s − 0.192·27-s + 1.28·29-s + 0.520·31-s − 0.0782·33-s − 0.414·35-s − 0.328·37-s − 1.10·39-s + 1.14·41-s − 1.96·43-s + 0.365·45-s − 1.29·47-s + 0.142·49-s − 0.217·51-s − 1.37·53-s + 0.148·55-s + ⋯

Functional equation

Λ(s)=(5376s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5376s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 53765376    =    28⋅3⋅72^{8} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 42.927542.9275
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5376, ( :1/2), 1)(2,\ 5376,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4664465562.466446556
L(12)L(\frac12) ≈\approx 2.4664465562.466446556
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}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
7 1+T 1 + T
good5 1−2.44T+5T2 1 - 2.44T + 5T^{2}
11 1−0.449T+11T2 1 - 0.449T + 11T^{2}
13 1−6.89T+13T2 1 - 6.89T + 13T^{2}
17 1−1.55T+17T2 1 - 1.55T + 17T^{2}
19 1−6.89T+19T2 1 - 6.89T + 19T^{2}
23 1−4.44T+23T2 1 - 4.44T + 23T^{2}
29 1−6.89T+29T2 1 - 6.89T + 29T^{2}
31 1−2.89T+31T2 1 - 2.89T + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1−7.34T+41T2 1 - 7.34T + 41T^{2}
43 1+12.8T+43T2 1 + 12.8T + 43T^{2}
47 1+8.89T+47T2 1 + 8.89T + 47T^{2}
53 1+10T+53T2 1 + 10T + 53T^{2}
59 1−4.89T+59T2 1 - 4.89T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+9.79T+67T2 1 + 9.79T + 67T^{2}
71 1−5.34T+71T2 1 - 5.34T + 71T^{2}
73 1+1.10T+73T2 1 + 1.10T + 73T^{2}
79 1+3.10T+79T2 1 + 3.10T + 79T^{2}
83 1+13.7T+83T2 1 + 13.7T + 83T^{2}
89 1+7.34T+89T2 1 + 7.34T + 89T^{2}
97 1−14.8T+97T2 1 - 14.8T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.303884433830111861966787166098, −7.30024699782449396220593784841, −6.45973483613623277103401740760, −6.12865365807960846870641605159, −5.39965968314420612940774502107, −4.72790362872200368450615652547, −3.54280303251370920413461316352, −2.96731371383366497259352898662, −1.57903243034526033198796369456, −0.982644963701242080047788468198, 0.982644963701242080047788468198, 1.57903243034526033198796369456, 2.96731371383366497259352898662, 3.54280303251370920413461316352, 4.72790362872200368450615652547, 5.39965968314420612940774502107, 6.12865365807960846870641605159, 6.45973483613623277103401740760, 7.30024699782449396220593784841, 8.303884433830111861966787166098

Graph of the ZZ-function along the critical line