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Properties

Label 2-1710-1.1-c1-0-6
Degree 22
Conductor 17101710
Sign 11
Analytic cond. 13.654413.6544
Root an. cond. 3.695183.69518
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 − 5-s − 3.12·7-s + 8-s − 10-s + 2·11-s + 4·13-s − 3.12·14-s + 16-s + 3.12·17-s − 19-s − 20-s + 2·22-s + 25-s + 4·26-s − 3.12·28-s − 2·29-s + 9.12·31-s + 32-s + 3.12·34-s + 3.12·35-s − 38-s − 40-s − 5.12·41-s + 10.2·43-s + 2·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.18·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s + 1.10·13-s − 0.834·14-s + 0.250·16-s + 0.757·17-s − 0.229·19-s − 0.223·20-s + 0.426·22-s + 0.200·25-s + 0.784·26-s − 0.590·28-s − 0.371·29-s + 1.63·31-s + 0.176·32-s + 0.535·34-s + 0.527·35-s − 0.162·38-s − 0.158·40-s − 0.800·41-s + 1.56·43-s + 0.301·44-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: 11
Analytic conductor: 13.654413.6544
Root analytic conductor: 3.695183.69518
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1710, ( :1/2), 1)(2,\ 1710,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4096439112.409643911
L(12)L(\frac12) ≈\approx 2.4096439112.409643911
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+T 1 + T
19 1+T 1 + T
good7 1+3.12T+7T2 1 + 3.12T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1−4T+13T2 1 - 4T + 13T^{2}
17 1−3.12T+17T2 1 - 3.12T + 17T^{2}
23 1+23T2 1 + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−9.12T+31T2 1 - 9.12T + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+5.12T+41T2 1 + 5.12T + 41T^{2}
43 1−10.2T+43T2 1 - 10.2T + 43T^{2}
47 1−10.2T+47T2 1 - 10.2T + 47T^{2}
53 1+4.24T+53T2 1 + 4.24T + 53T^{2}
59 1+3.12T+59T2 1 + 3.12T + 59T^{2}
61 1−12.2T+61T2 1 - 12.2T + 61T^{2}
67 1−6.24T+67T2 1 - 6.24T + 67T^{2}
71 1−6.24T+71T2 1 - 6.24T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−9.12T+79T2 1 - 9.12T + 79T^{2}
83 1−6.87T+83T2 1 - 6.87T + 83T^{2}
89 1+11.3T+89T2 1 + 11.3T + 89T^{2}
97 1+6T+97T2 1 + 6T + 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

−9.377066340388711656351793919491, −8.505170069146041669368657992845, −7.64656640993540486062457393481, −6.63283353310251304038819269910, −6.23941617932906518089671523408, −5.29048096807453615851225433327, −4.05195296127906493418242332683, −3.59658785251440597450313424611, −2.61640077576820863844940973511, −1.00477982265802441032029886721, 1.00477982265802441032029886721, 2.61640077576820863844940973511, 3.59658785251440597450313424611, 4.05195296127906493418242332683, 5.29048096807453615851225433327, 6.23941617932906518089671523408, 6.63283353310251304038819269910, 7.64656640993540486062457393481, 8.505170069146041669368657992845, 9.377066340388711656351793919491

Graph of the ZZ-function along the critical line