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Properties

Label 2-306-1.1-c1-0-1
Degree 22
Conductor 306306
Sign 11
Analytic cond. 2.443422.44342
Root an. cond. 1.563141.56314
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 4·11-s − 2·13-s + 16-s − 17-s + 4·19-s + 2·20-s − 4·22-s − 25-s + 2·26-s + 10·29-s + 8·31-s − 32-s + 34-s − 2·37-s − 4·38-s − 2·40-s − 10·41-s + 12·43-s + 4·44-s − 7·49-s + 50-s − 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 1.85·29-s + 1.43·31-s − 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.648·38-s − 0.316·40-s − 1.56·41-s + 1.82·43-s + 0.603·44-s − 49-s + 0.141·50-s − 0.277·52-s + ⋯

Functional equation

Λ(s)=(306s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(306s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 306306    =    2⋅32⋅172 \cdot 3^{2} \cdot 17
Sign: 11
Analytic conductor: 2.443422.44342
Root analytic conductor: 1.563141.56314
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 306, ( :1/2), 1)(2,\ 306,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1471740361.147174036
L(12)L(\frac12) ≈\approx 1.1471740361.147174036
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 1
17 1+T 1 + T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+12T+pT2 1 + 12 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+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−6T+pT2 1 - 6 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

−11.73531253806048906399243423876, −10.52566566553203306448675052964, −9.723242141956922614700830245165, −9.104444876573252449353403617805, −8.000937863938982294341146593643, −6.79533660929922109224513153399, −6.05886413503827459123865903257, −4.66119827871442481374467622682, −2.92323665285260202872977027719, −1.43438626728056627432143840048, 1.43438626728056627432143840048, 2.92323665285260202872977027719, 4.66119827871442481374467622682, 6.05886413503827459123865903257, 6.79533660929922109224513153399, 8.000937863938982294341146593643, 9.104444876573252449353403617805, 9.723242141956922614700830245165, 10.52566566553203306448675052964, 11.73531253806048906399243423876

Graph of the ZZ-function along the critical line