Properties

Label 2-2166-1.1-c3-0-25
Degree $2$
Conductor $2166$
Sign $1$
Analytic cond. $127.798$
Root an. cond. $11.3047$
Motivic weight $3$
Arithmetic yes
Rational yes
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 + 2·5-s − 6·6-s − 21·7-s + 8·8-s + 9·9-s + 4·10-s − 40·11-s − 12·12-s − 17·13-s − 42·14-s − 6·15-s + 16·16-s + 36·17-s + 18·18-s + 8·20-s + 63·21-s − 80·22-s + 74·23-s − 24·24-s − 121·25-s − 34·26-s − 27·27-s − 84·28-s − 100·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.178·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.126·10-s − 1.09·11-s − 0.288·12-s − 0.362·13-s − 0.801·14-s − 0.103·15-s + 1/4·16-s + 0.513·17-s + 0.235·18-s + 0.0894·20-s + 0.654·21-s − 0.775·22-s + 0.670·23-s − 0.204·24-s − 0.967·25-s − 0.256·26-s − 0.192·27-s − 0.566·28-s − 0.640·29-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.691519195\)
\(L(\frac12)\) \(\approx\) \(1.691519195\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 + p T \)
19 \( 1 \)
good5 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 + 3 p T + p^{3} T^{2} \)
11 \( 1 + 40 T + p^{3} T^{2} \)
13 \( 1 + 17 T + p^{3} T^{2} \)
17 \( 1 - 36 T + p^{3} T^{2} \)
23 \( 1 - 74 T + p^{3} T^{2} \)
29 \( 1 + 100 T + p^{3} T^{2} \)
31 \( 1 + 103 T + p^{3} T^{2} \)
37 \( 1 + 187 T + p^{3} T^{2} \)
41 \( 1 - 128 T + p^{3} T^{2} \)
43 \( 1 - 121 T + p^{3} T^{2} \)
47 \( 1 - 410 T + p^{3} T^{2} \)
53 \( 1 + 230 T + p^{3} T^{2} \)
59 \( 1 - 744 T + p^{3} T^{2} \)
61 \( 1 + 277 T + p^{3} T^{2} \)
67 \( 1 - 231 T + p^{3} T^{2} \)
71 \( 1 + 578 T + p^{3} T^{2} \)
73 \( 1 - 609 T + p^{3} T^{2} \)
79 \( 1 + 1259 T + p^{3} T^{2} \)
83 \( 1 + 696 T + p^{3} T^{2} \)
89 \( 1 - 612 T + p^{3} T^{2} \)
97 \( 1 - 1550 T + p^{3} T^{2} \)
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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

−8.775856108374878531449187699331, −7.53419908156586819715333775485, −7.15951993446566316224962620705, −6.07794227738289773583626943052, −5.63811354400338697591245895685, −4.85557505169767347573218618686, −3.79195687820374784759228949371, −2.99122520060694627347371736119, −2.01833480638905861877752155659, −0.51796278256806689093994961629, 0.51796278256806689093994961629, 2.01833480638905861877752155659, 2.99122520060694627347371736119, 3.79195687820374784759228949371, 4.85557505169767347573218618686, 5.63811354400338697591245895685, 6.07794227738289773583626943052, 7.15951993446566316224962620705, 7.53419908156586819715333775485, 8.775856108374878531449187699331

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