Properties

Label 2-2166-1.1-c3-0-122
Degree $2$
Conductor $2166$
Sign $1$
Analytic cond. $127.798$
Root an. cond. $11.3047$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s + 12.4·5-s + 6·6-s + 33.5·7-s + 8·8-s + 9·9-s + 24.9·10-s + 44.6·11-s + 12·12-s − 59.8·13-s + 67.0·14-s + 37.3·15-s + 16·16-s − 72.8·17-s + 18·18-s + 49.8·20-s + 100.·21-s + 89.2·22-s + 157.·23-s + 24·24-s + 30.1·25-s − 119.·26-s + 27·27-s + 134.·28-s + 16.2·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.11·5-s + 0.408·6-s + 1.81·7-s + 0.353·8-s + 0.333·9-s + 0.787·10-s + 1.22·11-s + 0.288·12-s − 1.27·13-s + 1.28·14-s + 0.643·15-s + 0.250·16-s − 1.03·17-s + 0.235·18-s + 0.557·20-s + 1.04·21-s + 0.865·22-s + 1.42·23-s + 0.204·24-s + 0.241·25-s − 0.903·26-s + 0.192·27-s + 0.905·28-s + 0.104·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2166\)    =    \(2 \cdot 3 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(127.798\)
Root analytic conductor: \(11.3047\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.745399159\)
\(L(\frac12)\) \(\approx\) \(7.745399159\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 - 3T \)
19 \( 1 \)
good5 \( 1 - 12.4T + 125T^{2} \)
7 \( 1 - 33.5T + 343T^{2} \)
11 \( 1 - 44.6T + 1.33e3T^{2} \)
13 \( 1 + 59.8T + 2.19e3T^{2} \)
17 \( 1 + 72.8T + 4.91e3T^{2} \)
23 \( 1 - 157.T + 1.21e4T^{2} \)
29 \( 1 - 16.2T + 2.43e4T^{2} \)
31 \( 1 + 265.T + 2.97e4T^{2} \)
37 \( 1 - 182.T + 5.06e4T^{2} \)
41 \( 1 - 147.T + 6.89e4T^{2} \)
43 \( 1 - 474.T + 7.95e4T^{2} \)
47 \( 1 + 331.T + 1.03e5T^{2} \)
53 \( 1 + 293.T + 1.48e5T^{2} \)
59 \( 1 + 256.T + 2.05e5T^{2} \)
61 \( 1 - 631.T + 2.26e5T^{2} \)
67 \( 1 - 170.T + 3.00e5T^{2} \)
71 \( 1 + 350.T + 3.57e5T^{2} \)
73 \( 1 + 915.T + 3.89e5T^{2} \)
79 \( 1 - 334.T + 4.93e5T^{2} \)
83 \( 1 - 731.T + 5.71e5T^{2} \)
89 \( 1 + 815.T + 7.04e5T^{2} \)
97 \( 1 - 1.20e3T + 9.12e5T^{2} \)
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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

−8.966791834033862909378655049921, −7.76772238163001528388153224795, −7.19355600823631156004195884830, −6.31028789310953207494908783476, −5.32580330435904081139639390645, −4.73021206957647948418094914009, −4.02311833693366235230496002353, −2.62645801795956667765121297013, −1.98128740973806870574953045505, −1.25025993648520812681789038341, 1.25025993648520812681789038341, 1.98128740973806870574953045505, 2.62645801795956667765121297013, 4.02311833693366235230496002353, 4.73021206957647948418094914009, 5.32580330435904081139639390645, 6.31028789310953207494908783476, 7.19355600823631156004195884830, 7.76772238163001528388153224795, 8.966791834033862909378655049921

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