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Properties

Label 2-966-1.1-c1-0-6
Degree 22
Conductor 966966
Sign 11
Analytic cond. 7.713547.71354
Root an. cond. 2.777322.77732
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 + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 6·11-s + 12-s + 2·13-s − 14-s + 16-s − 6·17-s − 18-s + 2·19-s + 21-s − 6·22-s + 23-s − 24-s − 5·25-s − 2·26-s + 27-s + 28-s − 6·29-s + 8·31-s − 32-s + 6·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s + 0.218·21-s − 1.27·22-s + 0.208·23-s − 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.04·33-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: 11
Analytic conductor: 7.713547.71354
Root analytic conductor: 2.777322.77732
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 966, ( :1/2), 1)(2,\ 966,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6331169361.633116936
L(12)L(\frac12) ≈\approx 1.6331169361.633116936
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−T 1 - T
23 1−T 1 - T
good5 1+pT2 1 + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−9.551879157003533670443999529649, −9.326791696013578560676280568052, −8.446718419123634224742538482321, −7.69912362113235286850265561333, −6.69924943503066031429063443018, −6.06776536207787590443613911254, −4.48140712011520856643820717997, −3.67993242300982530617744400082, −2.29445795919165996609896965567, −1.20369719133842088894251925117, 1.20369719133842088894251925117, 2.29445795919165996609896965567, 3.67993242300982530617744400082, 4.48140712011520856643820717997, 6.06776536207787590443613911254, 6.69924943503066031429063443018, 7.69912362113235286850265561333, 8.446718419123634224742538482321, 9.326791696013578560676280568052, 9.551879157003533670443999529649

Graph of the ZZ-function along the critical line