Properties

Label 2-2166-1.1-c1-0-34
Degree $2$
Conductor $2166$
Sign $1$
Analytic cond. $17.2955$
Root an. cond. $4.15879$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 1.18·5-s + 6-s + 0.532·7-s + 8-s + 9-s + 1.18·10-s + 1.87·11-s + 12-s + 3.87·13-s + 0.532·14-s + 1.18·15-s + 16-s + 1.16·17-s + 18-s + 1.18·20-s + 0.532·21-s + 1.87·22-s − 6.70·23-s + 24-s − 3.59·25-s + 3.87·26-s + 27-s + 0.532·28-s + 4.02·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.529·5-s + 0.408·6-s + 0.201·7-s + 0.353·8-s + 0.333·9-s + 0.374·10-s + 0.566·11-s + 0.288·12-s + 1.07·13-s + 0.142·14-s + 0.305·15-s + 0.250·16-s + 0.281·17-s + 0.235·18-s + 0.264·20-s + 0.116·21-s + 0.400·22-s − 1.39·23-s + 0.204·24-s − 0.719·25-s + 0.760·26-s + 0.192·27-s + 0.100·28-s + 0.746·29-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: no
Arithmetic: yes
Character: $\chi_{2166} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.203074721\)
\(L(\frac12)\) \(\approx\) \(4.203074721\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.18T + 5T^{2} \)
7 \( 1 - 0.532T + 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 - 3.87T + 13T^{2} \)
17 \( 1 - 1.16T + 17T^{2} \)
23 \( 1 + 6.70T + 23T^{2} \)
29 \( 1 - 4.02T + 29T^{2} \)
31 \( 1 - 1.95T + 31T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 - 8.98T + 41T^{2} \)
43 \( 1 - 2.42T + 43T^{2} \)
47 \( 1 - 2.04T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 2.68T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 6.07T + 71T^{2} \)
73 \( 1 + 0.327T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 3.55T + 89T^{2} \)
97 \( 1 + 5.87T + 97T^{2} \)
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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

−9.088762558639633628829018914732, −8.192821334243129047597978785512, −7.61269687974144945754820095390, −6.37538304332119573327121078952, −6.11253658685247986043717759249, −5.00341872275411380583833195329, −4.05698336335590967748236515580, −3.40556623100983644681039955843, −2.25886047239748081247111682439, −1.37465927691918900173775273371, 1.37465927691918900173775273371, 2.25886047239748081247111682439, 3.40556623100983644681039955843, 4.05698336335590967748236515580, 5.00341872275411380583833195329, 6.11253658685247986043717759249, 6.37538304332119573327121078952, 7.61269687974144945754820095390, 8.192821334243129047597978785512, 9.088762558639633628829018914732

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