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Properties

Label 2-8550-1.1-c1-0-14
Degree 22
Conductor 85508550
Sign 11
Analytic cond. 68.272068.2720
Root an. cond. 8.262698.26269
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  − 2-s + 4-s + 2.46·7-s − 8-s − 0.728·11-s − 6.23·13-s − 2.46·14-s + 16-s − 0.563·17-s − 19-s + 0.728·22-s − 4.63·23-s + 6.23·26-s + 2.46·28-s − 10.2·29-s + 6.06·31-s − 32-s + 0.563·34-s − 5.72·37-s + 38-s − 4.79·41-s + 8.06·43-s − 0.728·44-s + 4.63·46-s − 8.12·47-s − 0.900·49-s − 6.23·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.933·7-s − 0.353·8-s − 0.219·11-s − 1.72·13-s − 0.660·14-s + 0.250·16-s − 0.136·17-s − 0.229·19-s + 0.155·22-s − 0.966·23-s + 1.22·26-s + 0.466·28-s − 1.89·29-s + 1.08·31-s − 0.176·32-s + 0.0965·34-s − 0.941·37-s + 0.162·38-s − 0.748·41-s + 1.23·43-s − 0.109·44-s + 0.683·46-s − 1.18·47-s − 0.128·49-s − 0.864·52-s + ⋯

Functional equation

Λ(s)=(8550s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8550s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 85508550    =    2⋅32⋅52⋅192 \cdot 3^{2} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 68.272068.2720
Root analytic conductor: 8.262698.26269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8550, ( :1/2), 1)(2,\ 8550,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0398605841.039860584
L(12)L(\frac12) ≈\approx 1.0398605841.039860584
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+T 1 + T
3 1 1
5 1 1
19 1+T 1 + T
good7 1−2.46T+7T2 1 - 2.46T + 7T^{2}
11 1+0.728T+11T2 1 + 0.728T + 11T^{2}
13 1+6.23T+13T2 1 + 6.23T + 13T^{2}
17 1+0.563T+17T2 1 + 0.563T + 17T^{2}
23 1+4.63T+23T2 1 + 4.63T + 23T^{2}
29 1+10.2T+29T2 1 + 10.2T + 29T^{2}
31 1−6.06T+31T2 1 - 6.06T + 31T^{2}
37 1+5.72T+37T2 1 + 5.72T + 37T^{2}
41 1+4.79T+41T2 1 + 4.79T + 41T^{2}
43 1−8.06T+43T2 1 - 8.06T + 43T^{2}
47 1+8.12T+47T2 1 + 8.12T + 47T^{2}
53 1−1.53T+53T2 1 - 1.53T + 53T^{2}
59 1−5.76T+59T2 1 - 5.76T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1−12.9T+67T2 1 - 12.9T + 67T^{2}
71 1−4.39T+71T2 1 - 4.39T + 71T^{2}
73 1+4.09T+73T2 1 + 4.09T + 73T^{2}
79 1−15.3T+79T2 1 - 15.3T + 79T^{2}
83 1+7.85T+83T2 1 + 7.85T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−11.0T+97T2 1 - 11.0T + 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

−7.975239392201194734833546528512, −7.20711779628335621987261197324, −6.67131918531023650799606680141, −5.62402299051102079954756523583, −5.09002319492617793987502708246, −4.33982975164361804492220988575, −3.38425539607140498469565534189, −2.21501035721447891472946580252, −1.94696767510668984163891284021, −0.53093777693241158996295069653, 0.53093777693241158996295069653, 1.94696767510668984163891284021, 2.21501035721447891472946580252, 3.38425539607140498469565534189, 4.33982975164361804492220988575, 5.09002319492617793987502708246, 5.62402299051102079954756523583, 6.67131918531023650799606680141, 7.20711779628335621987261197324, 7.975239392201194734833546528512

Graph of the ZZ-function along the critical line