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Properties

Label 2-3822-1.1-c1-0-54
Degree 22
Conductor 38223822
Sign −1-1
Analytic cond. 30.518830.5188
Root an. cond. 5.524385.52438
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 − 3.56·5-s − 6-s + 8-s + 9-s − 3.56·10-s − 1.56·11-s − 12-s − 13-s + 3.56·15-s + 16-s + 6.68·17-s + 18-s + 4.68·19-s − 3.56·20-s − 1.56·22-s − 5.56·23-s − 24-s + 7.68·25-s − 26-s − 27-s + 6.68·29-s + 3.56·30-s − 6.24·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.59·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.12·10-s − 0.470·11-s − 0.288·12-s − 0.277·13-s + 0.919·15-s + 0.250·16-s + 1.62·17-s + 0.235·18-s + 1.07·19-s − 0.796·20-s − 0.332·22-s − 1.15·23-s − 0.204·24-s + 1.53·25-s − 0.196·26-s − 0.192·27-s + 1.24·29-s + 0.650·30-s − 1.12·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(3822s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3822s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38223822    =    2⋅3⋅72⋅132 \cdot 3 \cdot 7^{2} \cdot 13
Sign: −1-1
Analytic conductor: 30.518830.5188
Root analytic conductor: 5.524385.52438
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3822, ( :1/2), −1)(2,\ 3822,\ (\ :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
7 1 1
13 1+T 1 + T
good5 1+3.56T+5T2 1 + 3.56T + 5T^{2}
11 1+1.56T+11T2 1 + 1.56T + 11T^{2}
17 1−6.68T+17T2 1 - 6.68T + 17T^{2}
19 1−4.68T+19T2 1 - 4.68T + 19T^{2}
23 1+5.56T+23T2 1 + 5.56T + 23T^{2}
29 1−6.68T+29T2 1 - 6.68T + 29T^{2}
31 1+6.24T+31T2 1 + 6.24T + 31T^{2}
37 1+7.56T+37T2 1 + 7.56T + 37T^{2}
41 1−1.12T+41T2 1 - 1.12T + 41T^{2}
43 1+6.43T+43T2 1 + 6.43T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1+2.24T+59T2 1 + 2.24T + 59T^{2}
61 1+6.68T+61T2 1 + 6.68T + 61T^{2}
67 1+7.12T+67T2 1 + 7.12T + 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−3.56T+73T2 1 - 3.56T + 73T^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1+8.87T+83T2 1 + 8.87T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+14.4T+97T2 1 + 14.4T + 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

−7.83442632807584960067983105142, −7.42590511407383774850594312249, −6.72209315384959419675421368395, −5.58704493595776581039852349174, −5.19446594025614260546394525330, −4.23527472924228621408260472248, −3.59844997507620280971808870077, −2.86985591270793636190150429916, −1.31591471145023310684852207232, 0, 1.31591471145023310684852207232, 2.86985591270793636190150429916, 3.59844997507620280971808870077, 4.23527472924228621408260472248, 5.19446594025614260546394525330, 5.58704493595776581039852349174, 6.72209315384959419675421368395, 7.42590511407383774850594312249, 7.83442632807584960067983105142

Graph of the ZZ-function along the critical line