Properties

Label 2-1666-1.1-c1-0-31
Degree $2$
Conductor $1666$
Sign $1$
Analytic cond. $13.3030$
Root an. cond. $3.64733$
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 + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 6·11-s + 2·12-s − 2·13-s + 16-s + 17-s + 18-s + 4·19-s + 6·22-s + 2·24-s − 5·25-s − 2·26-s − 4·27-s + 4·31-s + 32-s + 12·33-s + 34-s + 36-s − 4·37-s + 4·38-s − 4·39-s − 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 1.27·22-s + 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s + 0.718·31-s + 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s − 0.657·37-s + 0.648·38-s − 0.640·39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1666\)    =    \(2 \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(13.3030\)
Root analytic conductor: \(3.64733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1666,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.227551659\)
\(L(\frac12)\) \(\approx\) \(4.227551659\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
17 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.435449454754526833910951528272, −8.550482821819457584609282080376, −7.75961744967946030738916282437, −6.98368217520104403156128027372, −6.16055468076817207109148001002, −5.16488039277883802975012518677, −4.04100934533934329070442787144, −3.50747685387131529534531471772, −2.52652883309992049334898398402, −1.44781063014612234378860428910, 1.44781063014612234378860428910, 2.52652883309992049334898398402, 3.50747685387131529534531471772, 4.04100934533934329070442787144, 5.16488039277883802975012518677, 6.16055468076817207109148001002, 6.98368217520104403156128027372, 7.75961744967946030738916282437, 8.550482821819457584609282080376, 9.435449454754526833910951528272

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