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Properties

Label 2-1666-17.16-c1-0-60
Degree $2$
Conductor $1666$
Sign $0.242 - 0.970i$
Analytic cond. $13.3030$
Root an. cond. $3.64733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3i·3-s + 4-s + 3i·6-s − 8-s − 6·9-s − 3i·11-s − 3i·12-s − 5·13-s + 16-s + (−1 + 4i)17-s + 6·18-s − 4·19-s + 3i·22-s − 4i·23-s + 3i·24-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73i·3-s + 0.5·4-s + 1.22i·6-s − 0.353·8-s − 2·9-s − 0.904i·11-s − 0.866i·12-s − 1.38·13-s + 0.250·16-s + (−0.242 + 0.970i)17-s + 1.41·18-s − 0.917·19-s + 0.639i·22-s − 0.834i·23-s + 0.612i·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1666\)    =    \(2 \cdot 7^{2} \cdot 17\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(13.3030\)
Root analytic conductor: \(3.64733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1666} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1666,\ (\ :1/2),\ 0.242 - 0.970i)\)

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 \( 1 + (1 - 4i)T \)
good3 \( 1 + 3iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 + iT - 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 7iT - 79T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.489383866212408429114831228512, −7.986510448782459581917771038150, −7.21853895521404731490139549341, −6.46255551782039909310330400563, −6.00244336362651258836625875751, −4.70006532541967167984193118624, −3.02899160385119718122384571544, −2.26966562329883433664405866920, −1.21550607021570668944503606432, 0, 2.22866957531329069298922253213, 3.12902590028434432809768617726, 4.36438747058234289352755295452, 4.84700614279977397693756689390, 5.77731796123063224794884499742, 7.00749818469806585289885518334, 7.67146978373313992190970942133, 8.830519550788509645442389591579, 9.358010237550467049694759648539

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