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Properties

Label 2-1338-1.1-c1-0-20
Degree 22
Conductor 13381338
Sign −1-1
Analytic cond. 10.683910.6839
Root an. cond. 3.268633.26863
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 0.326·5-s + 6-s − 1.59·7-s − 8-s + 9-s − 0.326·10-s − 2.21·11-s − 12-s + 2.49·13-s + 1.59·14-s − 0.326·15-s + 16-s + 6.46·17-s − 18-s − 7.66·19-s + 0.326·20-s + 1.59·21-s + 2.21·22-s + 2.91·23-s + 24-s − 4.89·25-s − 2.49·26-s − 27-s − 1.59·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.146·5-s + 0.408·6-s − 0.603·7-s − 0.353·8-s + 0.333·9-s − 0.103·10-s − 0.669·11-s − 0.288·12-s + 0.690·13-s + 0.426·14-s − 0.0842·15-s + 0.250·16-s + 1.56·17-s − 0.235·18-s − 1.75·19-s + 0.0730·20-s + 0.348·21-s + 0.473·22-s + 0.607·23-s + 0.204·24-s − 0.978·25-s − 0.488·26-s − 0.192·27-s − 0.301·28-s + ⋯

Functional equation

Λ(s)=(1338s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1338s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13381338    =    2⋅3⋅2232 \cdot 3 \cdot 223
Sign: −1-1
Analytic conductor: 10.683910.6839
Root analytic conductor: 3.268633.26863
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1338, ( :1/2), −1)(2,\ 1338,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
223 1+T 1 + T
good5 1−0.326T+5T2 1 - 0.326T + 5T^{2}
7 1+1.59T+7T2 1 + 1.59T + 7T^{2}
11 1+2.21T+11T2 1 + 2.21T + 11T^{2}
13 1−2.49T+13T2 1 - 2.49T + 13T^{2}
17 1−6.46T+17T2 1 - 6.46T + 17T^{2}
19 1+7.66T+19T2 1 + 7.66T + 19T^{2}
23 1−2.91T+23T2 1 - 2.91T + 23T^{2}
29 1+4.18T+29T2 1 + 4.18T + 29T^{2}
31 1−5.66T+31T2 1 - 5.66T + 31T^{2}
37 1+2.05T+37T2 1 + 2.05T + 37T^{2}
41 1−12.2T+41T2 1 - 12.2T + 41T^{2}
43 1+8.89T+43T2 1 + 8.89T + 43T^{2}
47 1+8.26T+47T2 1 + 8.26T + 47T^{2}
53 1+6.20T+53T2 1 + 6.20T + 53T^{2}
59 1+0.283T+59T2 1 + 0.283T + 59T^{2}
61 1+2.94T+61T2 1 + 2.94T + 61T^{2}
67 1+5.67T+67T2 1 + 5.67T + 67T^{2}
71 1−4.12T+71T2 1 - 4.12T + 71T^{2}
73 1−7.69T+73T2 1 - 7.69T + 73T^{2}
79 1+0.116T+79T2 1 + 0.116T + 79T^{2}
83 1+7.08T+83T2 1 + 7.08T + 83T^{2}
89 1−0.513T+89T2 1 - 0.513T + 89T^{2}
97 1+6.38T+97T2 1 + 6.38T + 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

−9.415982917934371246851115845873, −8.299396981918821509374762295242, −7.76442035004950498182218434377, −6.62974995463474002646012617052, −6.08909485588767440162569769402, −5.21752186008420517175535545674, −3.93312396850202973857380928848, −2.84122260867046052460528311042, −1.49242435063325155810800722200, 0, 1.49242435063325155810800722200, 2.84122260867046052460528311042, 3.93312396850202973857380928848, 5.21752186008420517175535545674, 6.08909485588767440162569769402, 6.62974995463474002646012617052, 7.76442035004950498182218434377, 8.299396981918821509374762295242, 9.415982917934371246851115845873

Graph of the ZZ-function along the critical line