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Properties

Label 2-5376-1.1-c1-0-33
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 + 2.73·5-s − 7-s + 9-s + 6.19·11-s − 2.73·15-s − 4.73·17-s + 0.535·19-s + 21-s + 5.26·23-s + 2.46·25-s − 27-s + 3.46·29-s + 8.92·31-s − 6.19·33-s − 2.73·35-s − 10·37-s − 4.73·41-s + 12.9·43-s + 2.73·45-s − 6.92·47-s + 49-s + 4.73·51-s + 3.46·53-s + 16.9·55-s − 0.535·57-s + 2.92·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.22·5-s − 0.377·7-s + 0.333·9-s + 1.86·11-s − 0.705·15-s − 1.14·17-s + 0.122·19-s + 0.218·21-s + 1.09·23-s + 0.492·25-s − 0.192·27-s + 0.643·29-s + 1.60·31-s − 1.07·33-s − 0.461·35-s − 1.64·37-s − 0.739·41-s + 1.97·43-s + 0.407·45-s − 1.01·47-s + 0.142·49-s + 0.662·51-s + 0.475·53-s + 2.28·55-s − 0.0709·57-s + 0.381·59-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 2.3169945732.316994573
L(12)L(\frac12) ≈\approx 2.3169945732.316994573
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 1−2.73T+5T2 1 - 2.73T + 5T^{2}
11 1−6.19T+11T2 1 - 6.19T + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+4.73T+17T2 1 + 4.73T + 17T^{2}
19 1−0.535T+19T2 1 - 0.535T + 19T^{2}
23 1−5.26T+23T2 1 - 5.26T + 23T^{2}
29 1−3.46T+29T2 1 - 3.46T + 29T^{2}
31 1−8.92T+31T2 1 - 8.92T + 31T^{2}
37 1+10T+37T2 1 + 10T + 37T^{2}
41 1+4.73T+41T2 1 + 4.73T + 41T^{2}
43 1−12.9T+43T2 1 - 12.9T + 43T^{2}
47 1+6.92T+47T2 1 + 6.92T + 47T^{2}
53 1−3.46T+53T2 1 - 3.46T + 53T^{2}
59 1−2.92T+59T2 1 - 2.92T + 59T^{2}
61 1−5.46T+61T2 1 - 5.46T + 61T^{2}
67 1+0.535T+67T2 1 + 0.535T + 67T^{2}
71 1−3.80T+71T2 1 - 3.80T + 71T^{2}
73 1−10.3T+73T2 1 - 10.3T + 73T^{2}
79 1−2.53T+79T2 1 - 2.53T + 79T^{2}
83 1+12.3T+83T2 1 + 12.3T + 83T^{2}
89 1−4.73T+89T2 1 - 4.73T + 89T^{2}
97 1+14.3T+97T2 1 + 14.3T + 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

−8.418127817264308191447454230740, −6.97043628321764978688023785594, −6.70509081619637486826052883013, −6.17416885115101496306723764960, −5.37308166137076779118576626552, −4.58667583527359396900236784806, −3.80418767014071736899528645652, −2.71992682841692787760051066851, −1.74307990653100790705285033155, −0.897847906683673217973501594445, 0.897847906683673217973501594445, 1.74307990653100790705285033155, 2.71992682841692787760051066851, 3.80418767014071736899528645652, 4.58667583527359396900236784806, 5.37308166137076779118576626552, 6.17416885115101496306723764960, 6.70509081619637486826052883013, 6.97043628321764978688023785594, 8.418127817264308191447454230740

Graph of the ZZ-function along the critical line