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Properties

Label 2-1666-1.1-c1-0-15
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s − 8-s + 9-s + 4·10-s − 4·11-s − 2·12-s + 4·13-s + 8·15-s + 16-s + 17-s − 18-s + 6·19-s − 4·20-s + 4·22-s + 2·24-s + 11·25-s − 4·26-s + 4·27-s + 6·29-s − 8·30-s − 4·31-s − 32-s + 8·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 0.577·12-s + 1.10·13-s + 2.06·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s − 0.894·20-s + 0.852·22-s + 0.408·24-s + 11/5·25-s − 0.784·26-s + 0.769·27-s + 1.11·29-s − 1.46·30-s − 0.718·31-s − 0.176·32-s + 1.39·33-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: \(1\)
Selberg data: \((2,\ 1666,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.513817616811344847287888964137, −8.385246104908393628644618576917, −7.30520133711344095212758347478, −6.86889957251780146284611049892, −5.61743213461748806697737608987, −5.04830930372003292108061612293, −3.81247805730279232139504007819, −2.98540643047270290280243464408, −1.02049582199051457089141407933, 0, 1.02049582199051457089141407933, 2.98540643047270290280243464408, 3.81247805730279232139504007819, 5.04830930372003292108061612293, 5.61743213461748806697737608987, 6.86889957251780146284611049892, 7.30520133711344095212758347478, 8.385246104908393628644618576917, 8.513817616811344847287888964137

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