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Properties

Label 2-1875-1.1-c1-0-28
Degree 22
Conductor 18751875
Sign 11
Analytic cond. 14.971914.9719
Root an. cond. 3.869363.86936
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  − 1.35·2-s − 3-s − 0.175·4-s + 1.35·6-s + 1.59·7-s + 2.93·8-s + 9-s + 3.33·11-s + 0.175·12-s + 7.05·13-s − 2.15·14-s − 3.61·16-s + 4.09·17-s − 1.35·18-s + 0.567·19-s − 1.59·21-s − 4.50·22-s + 6.30·23-s − 2.93·24-s − 9.52·26-s − 27-s − 0.279·28-s − 2.78·29-s − 0.995·31-s − 0.988·32-s − 3.33·33-s − 5.53·34-s + ⋯
L(s)  = 1  − 0.955·2-s − 0.577·3-s − 0.0876·4-s + 0.551·6-s + 0.603·7-s + 1.03·8-s + 0.333·9-s + 1.00·11-s + 0.0505·12-s + 1.95·13-s − 0.576·14-s − 0.904·16-s + 0.993·17-s − 0.318·18-s + 0.130·19-s − 0.348·21-s − 0.959·22-s + 1.31·23-s − 0.599·24-s − 1.86·26-s − 0.192·27-s − 0.0528·28-s − 0.516·29-s − 0.178·31-s − 0.174·32-s − 0.580·33-s − 0.948·34-s + ⋯

Functional equation

Λ(s)=(1875s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1875s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18751875    =    3⋅543 \cdot 5^{4}
Sign: 11
Analytic conductor: 14.971914.9719
Root analytic conductor: 3.869363.86936
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1875, ( :1/2), 1)(2,\ 1875,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0831153491.083115349
L(12)L(\frac12) ≈\approx 1.0831153491.083115349
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)
bad3 1+T 1 + T
5 1 1
good2 1+1.35T+2T2 1 + 1.35T + 2T^{2}
7 1−1.59T+7T2 1 - 1.59T + 7T^{2}
11 1−3.33T+11T2 1 - 3.33T + 11T^{2}
13 1−7.05T+13T2 1 - 7.05T + 13T^{2}
17 1−4.09T+17T2 1 - 4.09T + 17T^{2}
19 1−0.567T+19T2 1 - 0.567T + 19T^{2}
23 1−6.30T+23T2 1 - 6.30T + 23T^{2}
29 1+2.78T+29T2 1 + 2.78T + 29T^{2}
31 1+0.995T+31T2 1 + 0.995T + 31T^{2}
37 1+3.55T+37T2 1 + 3.55T + 37T^{2}
41 1−1.16T+41T2 1 - 1.16T + 41T^{2}
43 1−0.117T+43T2 1 - 0.117T + 43T^{2}
47 1−7.64T+47T2 1 - 7.64T + 47T^{2}
53 1−0.523T+53T2 1 - 0.523T + 53T^{2}
59 1−0.983T+59T2 1 - 0.983T + 59T^{2}
61 1−10.6T+61T2 1 - 10.6T + 61T^{2}
67 1+15.2T+67T2 1 + 15.2T + 67T^{2}
71 1+10.6T+71T2 1 + 10.6T + 71T^{2}
73 1−5.55T+73T2 1 - 5.55T + 73T^{2}
79 1+14.5T+79T2 1 + 14.5T + 79T^{2}
83 1−5.02T+83T2 1 - 5.02T + 83T^{2}
89 1−2.82T+89T2 1 - 2.82T + 89T^{2}
97 1+1.70T+97T2 1 + 1.70T + 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

−8.974202738068428167742934981653, −8.738805915098542627708178618202, −7.76025972263796058152352709409, −7.03923523733820074434378734471, −6.08358302792928319859791001929, −5.27690454012064343135306359959, −4.27464149691434021308747497577, −3.47464187506563361008315325574, −1.53941946295797567691624508938, −0.987353738408260084385826513754, 0.987353738408260084385826513754, 1.53941946295797567691624508938, 3.47464187506563361008315325574, 4.27464149691434021308747497577, 5.27690454012064343135306359959, 6.08358302792928319859791001929, 7.03923523733820074434378734471, 7.76025972263796058152352709409, 8.738805915098542627708178618202, 8.974202738068428167742934981653

Graph of the ZZ-function along the critical line