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Properties

Label 2-2254-1.1-c3-0-159
Degree 22
Conductor 22542254
Sign −1-1
Analytic cond. 132.990132.990
Root an. cond. 11.532111.5321
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 5.77·3-s + 4·4-s + 3.54·5-s − 11.5·6-s + 8·8-s + 6.31·9-s + 7.08·10-s − 6·11-s − 23.0·12-s − 21.4·13-s − 20.4·15-s + 16·16-s + 7.36·17-s + 12.6·18-s + 93.4·19-s + 14.1·20-s − 12·22-s + 23·23-s − 46.1·24-s − 112.·25-s − 42.9·26-s + 119.·27-s + 112.·29-s − 40.9·30-s − 286.·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.316·5-s − 0.785·6-s + 0.353·8-s + 0.233·9-s + 0.224·10-s − 0.164·11-s − 0.555·12-s − 0.458·13-s − 0.352·15-s + 0.250·16-s + 0.105·17-s + 0.165·18-s + 1.12·19-s + 0.158·20-s − 0.116·22-s + 0.208·23-s − 0.392·24-s − 0.899·25-s − 0.324·26-s + 0.850·27-s + 0.720·29-s − 0.248·30-s − 1.65·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(2254s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2254s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22542254    =    2⋅72⋅232 \cdot 7^{2} \cdot 23
Sign: −1-1
Analytic conductor: 132.990132.990
Root analytic conductor: 11.532111.5321
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2254, ( :3/2), −1)(2,\ 2254,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad2 1−2T 1 - 2T
7 1 1
23 1−23T 1 - 23T
good3 1+5.77T+27T2 1 + 5.77T + 27T^{2}
5 1−3.54T+125T2 1 - 3.54T + 125T^{2}
11 1+6T+1.33e3T2 1 + 6T + 1.33e3T^{2}
13 1+21.4T+2.19e3T2 1 + 21.4T + 2.19e3T^{2}
17 1−7.36T+4.91e3T2 1 - 7.36T + 4.91e3T^{2}
19 1−93.4T+6.85e3T2 1 - 93.4T + 6.85e3T^{2}
29 1−112.T+2.43e4T2 1 - 112.T + 2.43e4T^{2}
31 1+286.T+2.97e4T2 1 + 286.T + 2.97e4T^{2}
37 1+59.3T+5.06e4T2 1 + 59.3T + 5.06e4T^{2}
41 1+62.6T+6.89e4T2 1 + 62.6T + 6.89e4T^{2}
43 1−507.T+7.95e4T2 1 - 507.T + 7.95e4T^{2}
47 1+536.T+1.03e5T2 1 + 536.T + 1.03e5T^{2}
53 1−187.T+1.48e5T2 1 - 187.T + 1.48e5T^{2}
59 1+49.6T+2.05e5T2 1 + 49.6T + 2.05e5T^{2}
61 1−778.T+2.26e5T2 1 - 778.T + 2.26e5T^{2}
67 1+661.T+3.00e5T2 1 + 661.T + 3.00e5T^{2}
71 1−289.T+3.57e5T2 1 - 289.T + 3.57e5T^{2}
73 1−651.T+3.89e5T2 1 - 651.T + 3.89e5T^{2}
79 1+50.0T+4.93e5T2 1 + 50.0T + 4.93e5T^{2}
83 1+807.T+5.71e5T2 1 + 807.T + 5.71e5T^{2}
89 1+946.T+7.04e5T2 1 + 946.T + 7.04e5T^{2}
97 1+715.T+9.12e5T2 1 + 715.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

−8.090536348939550719534202170578, −7.23751385200936398460061222504, −6.57579829511820589130392500583, −5.54919293971300598348733980328, −5.44465835973939616448873339390, −4.45672545316847206766154441900, −3.41980830657220066875069720155, −2.40673171734908573687266624004, −1.21576741934918205902719968133, 0, 1.21576741934918205902719968133, 2.40673171734908573687266624004, 3.41980830657220066875069720155, 4.45672545316847206766154441900, 5.44465835973939616448873339390, 5.54919293971300598348733980328, 6.57579829511820589130392500583, 7.23751385200936398460061222504, 8.090536348939550719534202170578

Graph of the ZZ-function along the critical line