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Properties

Label 2-675-1.1-c3-0-25
Degree 22
Conductor 675675
Sign 11
Analytic cond. 39.826239.8262
Root an. cond. 6.310806.31080
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.53·2-s − 1.59·4-s + 27.8·7-s + 24.2·8-s + 35.8·11-s − 27.3·13-s − 70.5·14-s − 48.6·16-s + 93.6·17-s + 135.·19-s − 90.6·22-s − 0.407·23-s + 69.2·26-s − 44.5·28-s + 194.·29-s − 96.7·31-s − 71.1·32-s − 236.·34-s − 186.·37-s − 343.·38-s − 53.6·41-s − 519.·43-s − 57.2·44-s + 1.03·46-s + 190.·47-s + 434.·49-s + 43.7·52-s + ⋯
L(s)  = 1  − 0.894·2-s − 0.199·4-s + 1.50·7-s + 1.07·8-s + 0.981·11-s − 0.583·13-s − 1.34·14-s − 0.760·16-s + 1.33·17-s + 1.63·19-s − 0.878·22-s − 0.00369·23-s + 0.522·26-s − 0.300·28-s + 1.24·29-s − 0.560·31-s − 0.393·32-s − 1.19·34-s − 0.830·37-s − 1.46·38-s − 0.204·41-s − 1.84·43-s − 0.196·44-s + 0.00330·46-s + 0.591·47-s + 1.26·49-s + 0.116·52-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 39.826239.8262
Root analytic conductor: 6.310806.31080
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 675, ( :3/2), 1)(2,\ 675,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.5640552721.564055272
L(12)L(\frac12) ≈\approx 1.5640552721.564055272
L(52)L(\frac{5}{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 1
5 1 1
good2 1+2.53T+8T2 1 + 2.53T + 8T^{2}
7 1−27.8T+343T2 1 - 27.8T + 343T^{2}
11 1−35.8T+1.33e3T2 1 - 35.8T + 1.33e3T^{2}
13 1+27.3T+2.19e3T2 1 + 27.3T + 2.19e3T^{2}
17 1−93.6T+4.91e3T2 1 - 93.6T + 4.91e3T^{2}
19 1−135.T+6.85e3T2 1 - 135.T + 6.85e3T^{2}
23 1+0.407T+1.21e4T2 1 + 0.407T + 1.21e4T^{2}
29 1−194.T+2.43e4T2 1 - 194.T + 2.43e4T^{2}
31 1+96.7T+2.97e4T2 1 + 96.7T + 2.97e4T^{2}
37 1+186.T+5.06e4T2 1 + 186.T + 5.06e4T^{2}
41 1+53.6T+6.89e4T2 1 + 53.6T + 6.89e4T^{2}
43 1+519.T+7.95e4T2 1 + 519.T + 7.95e4T^{2}
47 1−190.T+1.03e5T2 1 - 190.T + 1.03e5T^{2}
53 1+533.T+1.48e5T2 1 + 533.T + 1.48e5T^{2}
59 1−472.T+2.05e5T2 1 - 472.T + 2.05e5T^{2}
61 1+327.T+2.26e5T2 1 + 327.T + 2.26e5T^{2}
67 1+78.0T+3.00e5T2 1 + 78.0T + 3.00e5T^{2}
71 1−707.T+3.57e5T2 1 - 707.T + 3.57e5T^{2}
73 1−344.T+3.89e5T2 1 - 344.T + 3.89e5T^{2}
79 1−98.3T+4.93e5T2 1 - 98.3T + 4.93e5T^{2}
83 1−1.24e3T+5.71e5T2 1 - 1.24e3T + 5.71e5T^{2}
89 1+448.T+7.04e5T2 1 + 448.T + 7.04e5T^{2}
97 1−472.T+9.12e5T2 1 - 472.T + 9.12e5T^{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

−9.920279182618299182715821057595, −9.280828050472990049248076615079, −8.297117857678951953725167394289, −7.78598324336757235089341246393, −6.92551060141358386486629757652, −5.33067488385657575166064099733, −4.76502419260906597287252224153, −3.48704778763113943636216315049, −1.69904924127376626688144351769, −0.936627922637311638810939463396, 0.936627922637311638810939463396, 1.69904924127376626688144351769, 3.48704778763113943636216315049, 4.76502419260906597287252224153, 5.33067488385657575166064099733, 6.92551060141358386486629757652, 7.78598324336757235089341246393, 8.297117857678951953725167394289, 9.280828050472990049248076615079, 9.920279182618299182715821057595

Graph of the ZZ-function along the critical line