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Properties

Label 2-1530-1.1-c1-0-17
Degree 22
Conductor 15301530
Sign −1-1
Analytic cond. 12.217112.2171
Root an. cond. 3.495293.49529
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 + 4-s − 5-s + 2·7-s − 8-s + 10-s − 4·13-s − 2·14-s + 16-s + 17-s − 4·19-s − 20-s + 25-s + 4·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s − 34-s − 2·35-s + 2·37-s + 4·38-s + 40-s + 2·43-s − 3·49-s − 50-s − 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.338·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.304·43-s − 3/7·49-s − 0.141·50-s − 0.554·52-s + ⋯

Functional equation

Λ(s)=(1530s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1530s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15301530    =    2⋅32⋅5⋅172 \cdot 3^{2} \cdot 5 \cdot 17
Sign: −1-1
Analytic conductor: 12.217112.2171
Root analytic conductor: 3.495293.49529
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1530, ( :1/2), −1)(2,\ 1530,\ (\ :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)Isogeny Class over Fp\mathbf{F}_p
bad2 1+T 1 + T
3 1 1
5 1+T 1 + T
17 1−T 1 - T
good7 1−2T+pT2 1 - 2 T + p T^{2} 1.7.ac
11 1+pT2 1 + p T^{2} 1.11.a
13 1+4T+pT2 1 + 4 T + p T^{2} 1.13.e
19 1+4T+pT2 1 + 4 T + p T^{2} 1.19.e
23 1+pT2 1 + p T^{2} 1.23.a
29 1+6T+pT2 1 + 6 T + p T^{2} 1.29.g
31 1+4T+pT2 1 + 4 T + p T^{2} 1.31.e
37 1−2T+pT2 1 - 2 T + p T^{2} 1.37.ac
41 1+pT2 1 + p T^{2} 1.41.a
43 1−2T+pT2 1 - 2 T + p T^{2} 1.43.ac
47 1+pT2 1 + p T^{2} 1.47.a
53 1−6T+pT2 1 - 6 T + p T^{2} 1.53.ag
59 1+6T+pT2 1 + 6 T + p T^{2} 1.59.g
61 1+10T+pT2 1 + 10 T + p T^{2} 1.61.k
67 1−2T+pT2 1 - 2 T + p T^{2} 1.67.ac
71 1−6T+pT2 1 - 6 T + p T^{2} 1.71.ag
73 1+16T+pT2 1 + 16 T + p T^{2} 1.73.q
79 1−8T+pT2 1 - 8 T + p T^{2} 1.79.ai
83 1+pT2 1 + p T^{2} 1.83.a
89 1+6T+pT2 1 + 6 T + p T^{2} 1.89.g
97 1+4T+pT2 1 + 4 T + p T^{2} 1.97.e
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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

−9.062768010948203852793619856758, −8.190885549450828256658813783292, −7.60443984278777067688587271846, −6.93439418815201615494528069017, −5.81883976600731049774743123462, −4.88219975867724665721913808894, −3.94620834878742004272258447586, −2.66245115412807667844263553225, −1.62441740500090931646073271114, 0, 1.62441740500090931646073271114, 2.66245115412807667844263553225, 3.94620834878742004272258447586, 4.88219975867724665721913808894, 5.81883976600731049774743123462, 6.93439418815201615494528069017, 7.60443984278777067688587271846, 8.190885549450828256658813783292, 9.062768010948203852793619856758

Graph of the ZZ-function along the critical line