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Properties

Label 2-110670-1.1-c1-0-39
Degree 22
Conductor 110670110670
Sign −1-1
Analytic cond. 883.704883.704
Root an. cond. 29.727129.7271
Motivic weight 11
Arithmetic yes
Rational yes
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 + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 2·11-s − 12-s + 4·13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 20-s − 21-s + 2·22-s − 6·23-s + 24-s + 25-s − 4·26-s − 27-s + 28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯

Functional equation

Λ(s)=(110670s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(110670s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110670110670    =    2⋅3⋅5⋅7⋅17⋅312 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 31
Sign: −1-1
Analytic conductor: 883.704883.704
Root analytic conductor: 29.727129.7271
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 110670, ( :1/2), −1)(2,\ 110670,\ (\ :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
5 1−T 1 - T
7 1−T 1 - T
17 1−T 1 - T
31 1−T 1 - T
good11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−4T+pT2 1 - 4 T + p T^{2}
97 1+14T+pT2 1 + 14 T + p T^{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

−13.90307722491801, −13.37431482274878, −12.90662118652067, −12.29044255944733, −11.83906967846240, −11.46058677971835, −10.74081784162253, −10.56617565684747, −10.01273218106002, −9.642284275021572, −8.862925545619899, −8.456661585952045, −8.062617008595254, −7.448096175106176, −6.903051237422671, −6.292455338078290, −5.872605044368580, −5.491403630874408, −4.702252240208072, −4.250388137687463, −3.398839462677633, −2.853642161303794, −2.010303676022876, −1.533192460572291, −0.8540721697947956, 0, 0.8540721697947956, 1.533192460572291, 2.010303676022876, 2.853642161303794, 3.398839462677633, 4.250388137687463, 4.702252240208072, 5.491403630874408, 5.872605044368580, 6.292455338078290, 6.903051237422671, 7.448096175106176, 8.062617008595254, 8.456661585952045, 8.862925545619899, 9.642284275021572, 10.01273218106002, 10.56617565684747, 10.74081784162253, 11.46058677971835, 11.83906967846240, 12.29044255944733, 12.90662118652067, 13.37431482274878, 13.90307722491801

Graph of the ZZ-function along the critical line