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Properties

Label 2-1254-1.1-c1-0-13
Degree 22
Conductor 12541254
Sign 11
Analytic cond. 10.013210.0132
Root an. cond. 3.164373.16437
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 − 3-s + 4-s + 3.18·5-s − 6-s + 0.508·7-s + 8-s + 9-s + 3.18·10-s + 11-s − 12-s + 0.508·14-s − 3.18·15-s + 16-s − 1.18·17-s + 18-s + 19-s + 3.18·20-s − 0.508·21-s + 22-s + 7.36·23-s − 24-s + 5.17·25-s − 27-s + 0.508·28-s − 9.53·29-s − 3.18·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.42·5-s − 0.408·6-s + 0.192·7-s + 0.353·8-s + 0.333·9-s + 1.00·10-s + 0.301·11-s − 0.288·12-s + 0.135·14-s − 0.823·15-s + 0.250·16-s − 0.288·17-s + 0.235·18-s + 0.229·19-s + 0.713·20-s − 0.110·21-s + 0.213·22-s + 1.53·23-s − 0.204·24-s + 1.03·25-s − 0.192·27-s + 0.0960·28-s − 1.77·29-s − 0.582·30-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), 1)(2,\ 1254,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9087991602.908799160
L(12)L(\frac12) ≈\approx 2.9087991602.908799160
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+T 1 + T
11 1−T 1 - T
19 1−T 1 - T
good5 1−3.18T+5T2 1 - 3.18T + 5T^{2}
7 1−0.508T+7T2 1 - 0.508T + 7T^{2}
13 1+13T2 1 + 13T^{2}
17 1+1.18T+17T2 1 + 1.18T + 17T^{2}
23 1−7.36T+23T2 1 - 7.36T + 23T^{2}
29 1+9.53T+29T2 1 + 9.53T + 29T^{2}
31 1−6.85T+31T2 1 - 6.85T + 31T^{2}
37 1−4.04T+37T2 1 - 4.04T + 37T^{2}
41 1+9.74T+41T2 1 + 9.74T + 41T^{2}
43 1+9.91T+43T2 1 + 9.91T + 43T^{2}
47 1−11.3T+47T2 1 - 11.3T + 47T^{2}
53 1+0.129T+53T2 1 + 0.129T + 53T^{2}
59 1+7.74T+59T2 1 + 7.74T + 59T^{2}
61 1+7.06T+61T2 1 + 7.06T + 61T^{2}
67 1−9.56T+67T2 1 - 9.56T + 67T^{2}
71 1−9.87T+71T2 1 - 9.87T + 71T^{2}
73 1−16.1T+73T2 1 - 16.1T + 73T^{2}
79 1−11.1T+79T2 1 - 11.1T + 79T^{2}
83 1−13.3T+83T2 1 - 13.3T + 83T^{2}
89 1−6.81T+89T2 1 - 6.81T + 89T^{2}
97 1+16.3T+97T2 1 + 16.3T + 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

−9.706204471945465236945291105467, −9.169857456620790870356461027297, −7.925036379224523473736107037620, −6.75764116923399700879048016389, −6.35639502830922980892388477456, −5.32136143288378522937095158200, −4.94286532816259896644578094023, −3.61006116920318506759304408828, −2.37845574161286393790094045573, −1.34864706644503212344032379864, 1.34864706644503212344032379864, 2.37845574161286393790094045573, 3.61006116920318506759304408828, 4.94286532816259896644578094023, 5.32136143288378522937095158200, 6.35639502830922980892388477456, 6.75764116923399700879048016389, 7.925036379224523473736107037620, 9.169857456620790870356461027297, 9.706204471945465236945291105467

Graph of the ZZ-function along the critical line