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Properties

Label 2-10-1.1-c15-0-1
Degree 22
Conductor 1010
Sign 11
Analytic cond. 14.269314.2693
Root an. cond. 3.777473.77747
Motivic weight 1515
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 128·2-s − 5.81e3·3-s + 1.63e4·4-s + 7.81e4·5-s − 7.44e5·6-s − 2.56e6·7-s + 2.09e6·8-s + 1.94e7·9-s + 1.00e7·10-s + 1.14e8·11-s − 9.52e7·12-s + 2.84e8·13-s − 3.28e8·14-s − 4.54e8·15-s + 2.68e8·16-s + 7.76e8·17-s + 2.49e9·18-s + 3.74e9·19-s + 1.28e9·20-s + 1.49e10·21-s + 1.46e10·22-s − 2.76e10·23-s − 1.21e10·24-s + 6.10e9·25-s + 3.64e10·26-s − 2.98e10·27-s − 4.19e10·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.53·3-s + 0.5·4-s + 0.447·5-s − 1.08·6-s − 1.17·7-s + 0.353·8-s + 1.35·9-s + 0.316·10-s + 1.77·11-s − 0.767·12-s + 1.25·13-s − 0.831·14-s − 0.686·15-s + 0.250·16-s + 0.459·17-s + 0.960·18-s + 0.959·19-s + 0.223·20-s + 1.80·21-s + 1.25·22-s − 1.69·23-s − 0.542·24-s + 0.200·25-s + 0.888·26-s − 0.549·27-s − 0.588·28-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=( Λ(16−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+15/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: 11
Analytic conductor: 14.269314.2693
Root analytic conductor: 3.777473.77747
Motivic weight: 1515
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 10, ( :15/2), 1)(2,\ 10,\ (\ :15/2),\ 1)

Particular Values

L(8)L(8) ≈\approx 1.8341040021.834104002
L(12)L(\frac12) ≈\approx 1.8341040021.834104002
L(172)L(\frac{17}{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 1−128T 1 - 128T
5 1−7.81e4T 1 - 7.81e4T
good3 1+5.81e3T+1.43e7T2 1 + 5.81e3T + 1.43e7T^{2}
7 1+2.56e6T+4.74e12T2 1 + 2.56e6T + 4.74e12T^{2}
11 1−1.14e8T+4.17e15T2 1 - 1.14e8T + 4.17e15T^{2}
13 1−2.84e8T+5.11e16T2 1 - 2.84e8T + 5.11e16T^{2}
17 1−7.76e8T+2.86e18T2 1 - 7.76e8T + 2.86e18T^{2}
19 1−3.74e9T+1.51e19T2 1 - 3.74e9T + 1.51e19T^{2}
23 1+2.76e10T+2.66e20T2 1 + 2.76e10T + 2.66e20T^{2}
29 1−1.59e10T+8.62e21T2 1 - 1.59e10T + 8.62e21T^{2}
31 1−2.79e9T+2.34e22T2 1 - 2.79e9T + 2.34e22T^{2}
37 1−6.85e11T+3.33e23T2 1 - 6.85e11T + 3.33e23T^{2}
41 1−9.09e11T+1.55e24T2 1 - 9.09e11T + 1.55e24T^{2}
43 1+4.11e11T+3.17e24T2 1 + 4.11e11T + 3.17e24T^{2}
47 1+4.29e11T+1.20e25T2 1 + 4.29e11T + 1.20e25T^{2}
53 1−1.11e13T+7.31e25T2 1 - 1.11e13T + 7.31e25T^{2}
59 1+2.98e13T+3.65e26T2 1 + 2.98e13T + 3.65e26T^{2}
61 1−1.56e13T+6.02e26T2 1 - 1.56e13T + 6.02e26T^{2}
67 1−6.47e13T+2.46e27T2 1 - 6.47e13T + 2.46e27T^{2}
71 1−1.38e14T+5.87e27T2 1 - 1.38e14T + 5.87e27T^{2}
73 1+5.76e13T+8.90e27T2 1 + 5.76e13T + 8.90e27T^{2}
79 1−1.85e14T+2.91e28T2 1 - 1.85e14T + 2.91e28T^{2}
83 1+3.07e14T+6.11e28T2 1 + 3.07e14T + 6.11e28T^{2}
89 1+1.81e14T+1.74e29T2 1 + 1.81e14T + 1.74e29T^{2}
97 1+1.82e14T+6.33e29T2 1 + 1.82e14T + 6.33e29T^{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

−16.77683200436030348164112878375, −16.01719778423616035572737560116, −13.92042686494785765624038293957, −12.43223860759449504643554059912, −11.40073356157614210382640980135, −9.777170891710757936159813134059, −6.51792103893196097256117846195, −5.87007184809743479809345747410, −3.84406424235310651912738150051, −1.05652376614415983558783477407, 1.05652376614415983558783477407, 3.84406424235310651912738150051, 5.87007184809743479809345747410, 6.51792103893196097256117846195, 9.777170891710757936159813134059, 11.40073356157614210382640980135, 12.43223860759449504643554059912, 13.92042686494785765624038293957, 16.01719778423616035572737560116, 16.77683200436030348164112878375

Graph of the ZZ-function along the critical line