Properties

Label 2-2166-1.1-c1-0-36
Degree $2$
Conductor $2166$
Sign $1$
Analytic cond. $17.2955$
Root an. cond. $4.15879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s + 4·11-s + 12-s + 4·14-s + 16-s − 2·17-s + 18-s + 4·21-s + 4·22-s − 2·23-s + 24-s − 5·25-s + 27-s + 4·28-s + 6·29-s − 6·31-s + 32-s + 4·33-s − 2·34-s + 36-s + 8·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.872·21-s + 0.852·22-s − 0.417·23-s + 0.204·24-s − 25-s + 0.192·27-s + 0.755·28-s + 1.11·29-s − 1.07·31-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2166\)    =    \(2 \cdot 3 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(17.2955\)
Root analytic conductor: \(4.15879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.312400268\)
\(L(\frac12)\) \(\approx\) \(4.312400268\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
19 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.891315695612574872146287348962, −8.266538915527224263777195540327, −7.54953259660607698064814994541, −6.72056041985753664693998194219, −5.86279269185425347466345430550, −4.82624966188247411001253938074, −4.26615040210543801862743429615, −3.41899742877711894209698559306, −2.14151289942301727352812099402, −1.44213087893724210061243055494, 1.44213087893724210061243055494, 2.14151289942301727352812099402, 3.41899742877711894209698559306, 4.26615040210543801862743429615, 4.82624966188247411001253938074, 5.86279269185425347466345430550, 6.72056041985753664693998194219, 7.54953259660607698064814994541, 8.266538915527224263777195540327, 8.891315695612574872146287348962

Graph of the $Z$-function along the critical line