Properties

Label 2-2166-1.1-c3-0-11
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 − 7·5-s − 6·6-s − 15·7-s + 8·8-s + 9·9-s − 14·10-s − 49·11-s − 12·12-s − 14·13-s − 30·14-s + 21·15-s + 16·16-s − 33·17-s + 18·18-s − 28·20-s + 45·21-s − 98·22-s − 148·23-s − 24·24-s − 76·25-s − 28·26-s − 27·27-s − 60·28-s + 278·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.626·5-s − 0.408·6-s − 0.809·7-s + 0.353·8-s + 1/3·9-s − 0.442·10-s − 1.34·11-s − 0.288·12-s − 0.298·13-s − 0.572·14-s + 0.361·15-s + 1/4·16-s − 0.470·17-s + 0.235·18-s − 0.313·20-s + 0.467·21-s − 0.949·22-s − 1.34·23-s − 0.204·24-s − 0.607·25-s − 0.211·26-s − 0.192·27-s − 0.404·28-s + 1.78·29-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: $\chi_{2166} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8403497874\)
\(L(\frac12)\) \(\approx\) \(0.8403497874\)
\(L(\frac{5}{2})\) 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 \( 1 + 7 T + p^{3} T^{2} \)
7 \( 1 + 15 T + p^{3} T^{2} \)
11 \( 1 + 49 T + p^{3} T^{2} \)
13 \( 1 + 14 T + p^{3} T^{2} \)
17 \( 1 + 33 T + p^{3} T^{2} \)
23 \( 1 + 148 T + p^{3} T^{2} \)
29 \( 1 - 278 T + p^{3} T^{2} \)
31 \( 1 + 94 T + p^{3} T^{2} \)
37 \( 1 + 160 T + p^{3} T^{2} \)
41 \( 1 + 400 T + p^{3} T^{2} \)
43 \( 1 - 73 T + p^{3} T^{2} \)
47 \( 1 - 173 T + p^{3} T^{2} \)
53 \( 1 + 170 T + p^{3} T^{2} \)
59 \( 1 - 12 T + p^{3} T^{2} \)
61 \( 1 - 419 T + p^{3} T^{2} \)
67 \( 1 + 444 T + p^{3} T^{2} \)
71 \( 1 - 952 T + p^{3} T^{2} \)
73 \( 1 + 27 T + p^{3} T^{2} \)
79 \( 1 - 556 T + p^{3} T^{2} \)
83 \( 1 + 276 T + p^{3} T^{2} \)
89 \( 1 + 1386 T + p^{3} T^{2} \)
97 \( 1 + 130 T + p^{3} T^{2} \)
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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

−8.489339103197148158185701807466, −7.82200510857480130095675491104, −6.98702407686584684973630990008, −6.31255153739937207197465657630, −5.47919986680596515476883418946, −4.74979932933948890875611834490, −3.88891588361198760084411014626, −3.01377406472305668177120501580, −2.02924033501782736534904015036, −0.35911169289529144392631994048, 0.35911169289529144392631994048, 2.02924033501782736534904015036, 3.01377406472305668177120501580, 3.88891588361198760084411014626, 4.74979932933948890875611834490, 5.47919986680596515476883418946, 6.31255153739937207197465657630, 6.98702407686584684973630990008, 7.82200510857480130095675491104, 8.489339103197148158185701807466

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